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# Transitional dynamics of the search model with endogenous growth

Fabien Postel-Vinay

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## To cite this version:

Fabien Postel-Vinay. Transitional dynamics of the search model with endogenous growth. 1996. halshs-05087402

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CAHIERS ECO & MATHS

96.46

TRANSITIONAL DYNAMICS OF THE SEARCH MODEL
WITH ENDOGENOUS GROWTH

Fabien POSTEL-VINAY

MAD
TRANSITIONAL DYNAMICS OF THE SEARCH MODEL WITH
ENDOGENOUS GROWTH

Fabien Postel-Vinay

MAD, Université de Paris I*†

Summer' 96 - First Shot

ABSTRACT

This paper studies the steady state and dynamics of the standard matching model, allowing for endogenous capital accumulation and endogenous growth. The existence and uniqueness of a balanced growth path is shown, along which the relationship between unemployment and growth may not be considered clear-cut. The model also exhibits interesting dynamic features, and suggests some kind of asymmetry in the adjustment of employment over the cycle. A collection of numerical experiments finally shows that a linear approximation of the model may yield a dynamic scheme that is reasonably close to that obtained with the nonlinear model, and that this feature is very robust to changes in particular assumptions made on the specification of various functions.

**Key words:** endogenous growth, unemployment, matching.

**JEL classification:** E24, J41, O41.

RÉSUMÉ

Cet article étudie l'état stationnaire et la dynamique du modèle standard d'appariement sur le marché du travail avec accumulation de capital et croissance endogènes. Il démontre l'existence et l'unicité d'un sentier de croissance équilibrée, le long duquel la relation entre taux de chômage et taux de croissance peut être ambiguë. Le modèle possède en outre d'intéressantes propriétés dynamiques, et suggère que l'ajustement du niveau de l'emploi au cours du cycle est asymétrique. Une série d'expériences numériques montre enfin qu'une approximation linéaire du modèle peut produire une dynamique raisonablement proche de celle du modèle non-linéaire, et que cette caractéristique est remarquablement robuste aux changements de spécifications.

**Mots-clefs :** croissance engogène, chômage, appariement.

<footnote>*Address: MAD, CNRS-URA 0926, Université de Paris I, 90 rue de Tolbiac 75634 Paris Cedex 13, France.
E-mail: fpostel@univ-paris1.fr</footnote>

<footnote>†I wish to thank A. d'Autume, P. Cahuc, P. Fève, E. Lehmann, M. Pucci and J.-M. Tallon for their careful reading of this paper, and for their valuable suggestions. All remaining errors are my own.</footnote>
# I. INTRODUCTION

The recently expanding literature on the link between unemployment and endogenous growth mainly focuses on the study of the comparative static properties of rich dynamic models. The study of transitional dynamics remains, to a large extent, a blank in this literature (although the pioneering work of Ito [1980] may be considered a most remarkable exception – with exogenous growth, though). This article intends to take a small step toward the filling of that blank by studying the dynamics of the well-known Pissarides [1990] search model in an endogenous growth framework à la Romer [1986].

The steady states of such an economy were analyzed by Bean and Pissarides [1993] in an overlapping generations framework. Beside that, the dynamics of the search model are studied in Blanchard and Diamond [1989], Pissarides [1985], Pissarides [1987], and Pissarides [1990], chapter 3 under the simplifying assumptions of a constant output per employee, an exogenous demand side and no capital accumulation. Mortensen [1989] drops the assumption of constant returns to labor, and Mortensen [1994] takes the possibility of job-to job mobility into account, still omitting capital. Finally, the introduction of endogenous capital accumulation in search models has been carried out in recent contributions to the literature on the so-called Real Business Cycle models (Fève and Langot [1996], Merz [1995], Langot [1995], Langot [1996]). As for the model herein, the basic input of RBC models is neoclassical growth. However, their main purpose is to mimic aggregate time series data, an exercise in which they perform pretty well. And due to the general lack of tractability of RBC models, the dynamics are obtained by calibrating and simulating a linearized version of their system of dynamic equations. Thus, yet those models do well in capturing most of the stylized facts that are known about the behavior of aggregate variables, they remain somewhat unsatisfactory from a theoretical viewpoint. Moreover, they still handle growth as an exogenous phenomenon.

In this article, we take the standard search model of Pissarides [1990] and allow for endogenous capital accumulation and for unbounded endogenous growth by means of a production externality implying non-decreasing returns to capital. We show the existence and uniqueness of the balanced growth path of the model. Some comparative static exercises suggest that the sign of the relationship between the long-run rate of growth and the steady state level of employment is not totally clear-cut. We then study the dynamic adjustment of the variables toward the steady state. This adjustment is illustrated in a gathering of numerical experiments. We
show that the dynamics of the complete model are, in a number of cases, well approximated
by the simpler models studied in some of the contributions cited above, despite the presence of
capital. Furthermore, we argue that a linearized version of the model should also yield a very
reasonable proxy for the dynamics of the actual nonlinear model, and that this feature that
we term ‘quasi-linearity’ is remarkably robust to changes in specifications. We also point out
important dynamic features of the model that are due to both endogenous capital accumulation
and endogenous growth. One is the traditional hysteretic nature of Romer-style endogenous
growth models, which is of particular interest in an economy experiencing underemployment,
since unemployment is an endogenous cause of low savings leading to slow accumulation and
slow growth. Another one deals with the question of unemployment persistence: the model
suggests that underemployment is more persistent than overemployment, a result that is well-
known to be consistent altogether with Keynes’s thought about business cycles (Keynes [1936],
book VI, chapter 22), the available time series data (e.g. Neftçi [1984], DeLong and Summers
[1986]), and the theoretical findings of alternative approaches to labor market analysis, such as
the *insider-outsider* theory of unemployment.

Section II presents the basic framework. Section III studies the balanced growth path and the comparative static properties of the model. Section IV carries out the study of its dynamics. It also shows the results of some simulations of the model economy. Section V is a remake of sections III and IV under a different assumption made about the cost structure, and provides an indication of the robustness of the results to changes in assumptions made about functional forms. Section VI concludes.

## II. THE BASIC SETUP

This section merely describes the basic framework we will be working with. The economy we
consider is basically the same as that of Pissarides [1990]'s second chapter. Time is continuous,
and all variables are indexed by time. All agents have perfect foresight. The first subsection
handles the goods market, and the second one deals with the labor market.

### II.1. *Production, consumption, accumulation, and goods market equilibrium*

One good is produced by a large (fixed) number of identical firms, using labor and capital
as inputs. This good can either be consumed by households or accumulated. The production
technology is the following:

$$y_t = f(k_t, \bar{k}_t \ell_t), \qquad (1)$$

where $y_t$ is the representative firm's output, $k_t$ the stock of capital in that firm, and $\ell_t$ the quantity of labor it employs. $\bar{k}_t$ denotes the aggregate stock of capital (a bar over a variable indicates an aggregate quantity, taken as given by individual), which appears here as a Romer-style externality to make unbounded endogenous growth possible. The production function is of the standard neoclassical variety.

The next assumption we make is that the labor market undergoes some frictions, so that if at some date $t$ the number of unemployed workers in the economy is $\bar{u}_t$ and the number of vacant jobs is $\bar{v}_t$, the actual number of hires at date $t$ is $m(\bar{u}_t, \bar{v}_t) \le \min\{\bar{u}_t, \bar{v}_t\}$, where the matching technology $m$ exhibits constant returns to scale, together with the ensuing additional properties: $m(\bar{u}_t, 0) = m(0, \bar{v}_t) = 0$, $\lim_{\bar{u}_t \to +\infty} m(\bar{u}_t, \bar{v}_t) = \bar{v}_t$, and $\lim_{\bar{v}_t \to +\infty} m(\bar{u}_t, \bar{v}_t) = \bar{u}_t$. We next define the usual tightness parameter $\bar{\theta}_t = \bar{v}_t/\bar{u}_t$, and the hiring rate $q(\bar{\theta}_t) = m(1, \bar{\theta}_t)/\bar{\theta}_t$. A given firm's hires are proportional to the number $v_t$ of vacancies it has, and is thus given by $q(\bar{\theta}_t) \cdot v_t$. Jobs are terminated at some exogenous rate of 'structural change' $\mu$, so that employment in the representative firm evolves according to:

$$\dot{\ell}_t = q(\bar{\theta}_t) \cdot v_t - \mu \ell_t. \qquad (2)$$

A vacancy has a real cost of $\kappa_t$ per unit time. Capital does not depreciate. Firms maximize the present discounted value of their expected profits, hence solving:

$$\max_{\{i_t, v_t\}} \int_0^{+\infty} e^{-\int_0^t r_\tau d\tau} \cdot [f(k_t, \bar{k}_t \ell_t) - w_t \ell_t - \kappa_t v_t - i_t] dt \qquad (3)$$

$$\text{subject to:} \quad \begin{cases} \dot{\ell}_t = q(\bar{\theta}_t) \cdot v_t - \mu \ell_t \\ \dot{\bar{k}}_t = i_t \\ k_0 \text{ and } \ell_0 \text{ given.} \end{cases}$$

where $i_t$ is investment at date $t$, $r_t$ the interest rate, and $w_t$ the wage. Firms take interest rates, wages, and hiring costs as given, as well as the path of the aggregate capital stock $\bar{k}_t$, and the tightness parameter $\bar{\theta}_t$. Note that employment is not a control variable of the firms, due to the imperfection of the matching technology. Using Pontryagin's principle, and denoting by $\pi_t$ the shadow price of an occupied job, one gets the optimality conditions for problem (3):

$$f'_1(k_t, \bar{k}_t \ell_t) = r_t, \qquad (4)$$

$$\pi_t = \frac{\kappa_t}{q(\bar{\theta}_t)}, \qquad (5)$$
$$r_t \pi_t = k_t f_2'(k_t, \bar{k}_t \ell_t) - w_t - \mu \pi_t + \dot{\pi}_t, \quad (6)$$

$$\lim_{t \to +\infty} e^{-\int_0^t r_\tau d\tau} \cdot \pi_t \ell_t = 0 \quad \text{and} \quad \lim_{t \to +\infty} e^{-\int_0^t r_\tau d\tau} \cdot k_t = 0, \quad (7)$$

(the two equations in (7) being transversality conditions). Firms equalize the cost of a vacancy ($\kappa_t$) to its expected return $q(\bar{\theta}_t) \pi_t$, which is also the shadow price of a vacant job slot. The evolution of that shadow price is then determined by the arbitrage equation (6). This arbitrage equation is the key to the dynamic adjustment of the number of vacancies. To see this, let us replace (4) and (5) into (6) and normalize the size of the economy so that, ex post, all aggregate variables become equal to their individual counterparts. This allows us to omit the bars in equilibrium, what we shall do from now on. One gets the dynamic equation for the tightness parameter:

$$-\eta_{\theta,t}^q \cdot \frac{\dot{\theta}_t}{\theta_t} = f_1'(1, \ell_t) + \mu - \frac{q(\theta_t)}{\kappa_t} \cdot \{k_t f_2'(1, \ell_t) - w_t\} - \frac{\dot{\kappa}_t}{\kappa_t}, \quad (8)$$

with $\eta_{\theta,t}^q < 0$ denoting the elasticity of the hiring rate $q$ w.r.t. its argument, $\theta_t$ (note that this elasticity is *not* assumed constant).

Finally, assume that labor is supplied inelastically, and that the amount of labor available is normalized to one<fnref n="1" />. This normalization implies that $u_t = 1 - \ell_t$. This allows us to rewrite equation (2) in terms of $(\theta_t, \ell_t)$:

$$\dot{\ell}_t = \theta_t q(\theta_t) \cdot (1 - \ell_t) - \mu \ell_t. \quad (9)$$

We now turn to households.

They supply labor without disutility. They work, consume and lend capital to the firms (financial markets are kept implicit). They are endowed with a CIES felicity function, with intertemporal elasticity of substitution $\sigma$, and a discount factor $\rho$. If $c_t$ is consumption, it evolves according to the Keynes-Ramsey rule<fnref n="2" />:

$$\frac{\dot{c}_t}{c_t} = \sigma (r_t - \rho), \quad (10)$$

and satisfies the no-Ponzi game condition:

$$\lim_{t \to +\infty} e^{-\int_0^t r_\tau d\tau} \cdot k_t = 0, \quad (11)$$

<footnote n="1">¹We shall assume throughout the paper that labor supply (i.e. 1) never falls short of labor demand, so that actual labor is $\ell_t$.</footnote>

<footnote n="2">²It is implicitly assumed here that worker households can insure themselves against labor market uncertainties, so that their instantaneous budget constraints can be considered deterministic. Another option, that is adopted by Merz [1995], is to think of worker households as large families with a continuum of members insuring each other against labor income variations.</footnote>
which is the same as the transversality condition of the firms' programme for the stock of capital (the second condition in (7)). Finally, equilibrium on the goods market is written (ex post) as:

$$ \dot{k}_t = k_t f(1, \ell_t) - c_t. \qquad (12) $$

For the remainder, it will be convenient to consider the detrended control-like variable $\chi_t = c_t/k_t$ (the consumption-capital ratio). From (10) and (12), one sees that $\chi_t$ solves the ensuing equation:

$$ \frac{\dot{\chi}_t}{\chi_t} = \sigma [f'_1(1, \ell_t) - \rho] - f(1, \ell_t) + \chi_t. \qquad (13) $$

We now have a dynamic system describing our economy consisting of equations (8), (9), and (13). This system will be referred to as (S1) in the remainder. It involves two forward looking variables (namely $\chi_t$ and $\theta_t$), and one predetermined variable ($\ell_t$).

The status of the hiring cost $\kappa_t$, as that of the wage $w_t$, has remained unclear until now. About the former, a reasonable hypothesis is to make it proportional to the wage, implicitly assuming that hiring is a labor intensive activity (see e.g. chapter 2 of Pissarides [1990])<fnref n="3" />. We thus introduce the constant parameter $\kappa > 0$ such that $\kappa_t = \kappa w_t$. We now have to be more precise about the modelling of wage formation, for which there are several possibilities available (see Mortensen [1989]). Although imperfect matching is a source of nontrivial unemployment by itself, we shall assume here that wages are the result of a continuous decentralized bargain between the firm and its marginal worker, which is an additional cause of inefficient underemployment. The argument to justify the existence of this bargain is simply that a successful match generates some rent for the entrepreneur-worker pair that has to be shared. The next subsection describes the way this sharing is achieved.

## II.2. The wage bargain

Let $\nu \in (0, 1)$ be the relative bargaining power of the worker<fnref n="4" />. The outcome of the bargain is classically supposed to split the global surplus into two shares that have a relative size of $\nu/(1-\nu)$, that is:

$$ \Omega_t^e - \bar{\Omega}_t^u = \frac{\nu}{1-\nu} \cdot (\Pi_t^o - \bar{\Pi}_t^v), \qquad (14) $$

<footnote n="3">³Another option – that is chosen by Bean and Pissarides [1993] – is to make it follow the state of technical progress. Available interpretations for this assumption (see the article just cited) are somewhat less convincing. Furthermore, it appears that both options are equivalent in the long run, albeit differing in their influences on the short run dynamics (see the discussion in the beginning of section V below).</footnote>

<footnote n="4">⁴To ensure that workers and firms have different target functions in the negotiation, we assume that households are divided into two distinct categories: rentiers, to whom the dividends are distributed and who do not work, and workers. Both categories have similar preferences.</footnote>
with $\Omega_t^e$ denoting the (non financial) wealth of an employed worker in a given firm at some date $t$, $\overline{\Omega}_t^u$ that of an unemployed one$^5$, $\Pi_t^o$ the value of a given firm's marginal (occupied) job, and $\overline{\Pi}_t^v$ the value of a vacant job in that firm. The value functions defined above may be derived in the following way: first, $\Omega_t^e$ evolves according to the ensuing asset equation:

$$r_t \Omega_t^e = w_t - b_t + \dot{\Omega}_t^e - \mu (\Omega_t^e - \overline{\Omega}_t^u), \quad (15)$$

with $b_t$ a lump-sum tax meant to finance unemployment subsidies (if any). Similarly, $\theta_t q(\theta_t)$ is the instantaneous probability for an unemployed worker to match a vacant job. We denote by $z_t$ the (gross) per period earnings of an unemployed worker. It may solely consist of unemployment benefits, or reflect in a more general sense the returns to homework and the valuation of leisure time. We will return to the meaning of $z_t$ in a while. Assuming that the tax is paid by the unemployed as well, one has:

$$r_t \overline{\Omega}_t^u = z_t - b_t + \dot{\overline{\Omega}}_t^u - \theta_t q(\theta_t) \cdot (\overline{\Omega}_t^u - \overline{\Omega}_t^e), \quad (16)$$

with $\overline{\Omega}_t^e$ denoting the average wealth of an employed worker, thus taken as given by individuals.

Next, we know that a vacant job costs $\kappa_t$ and returns $q(\theta_t)$ workers per unit time. Consequently, the asset equation of $\Pi_t^v$ can be written as:

$$r_t \overline{\Pi}_t^v = -\kappa_t + \dot{\overline{\Pi}}_t^v - q(\theta_t) \cdot (\overline{\Pi}_t^v - \Pi_t^o). \quad (17)$$

Finally, assume that firms are free to advertise for as many job slots as they like, so that they exploit all profit opportunities from opening such job slots. Henceforth one has $\forall t, \overline{\Pi}_t^v = \dot{\overline{\Pi}}_t^v = 0$. The latter asset equation implies that an occupied job is valued (ex post) at its shadow price $\pi_t$:

$$\Pi_t^o = \pi_t = \frac{\kappa_t}{q(\theta_t)}. \quad (18)$$

Apart from that, the value to the firms of an occupied job evolves according to the shadow price equation (6):

$$r_t \Pi_t^o = k_t f_2'(1, \ell_t) - w_t + \dot{\Pi}_t^o - \mu \Pi_t^o, \quad (19)$$

Now operating a standard mixing$^6$ of equations (14) through (19), one gets the wage equation:

$$w_t = (1 - \nu) z_t + \nu [\theta_t \cdot \kappa_t + k_t f_2'(1, \ell_t)]. \quad (20)$$

<footnote>$^5$A bar over a variable is still used to denote an average value, taken as given by individuals, in contrast with individual values (those without bars).</footnote>

<footnote>$^6$See e.g. Pissarides [1990], chapter 2.</footnote>
The lump-sum tax assumption implies that $b_t$ vanishes from the wage equation, a convenient feature. Interpreting this equation is an easy task: the reward of an employed worker equals a weighted average of his alternative instantaneous gain, $z_t$, and the instantaneous profit he generates (that profit includes the saving on hiring costs: $\theta_t \cdot \kappa_t$ is the average hiring cost for each unemployed worker). Note that equation (20) includes the special case of a 'competitive' wage equation, corresponding to zero weight for the workers in the negotiation ($\nu = 0, w_t = z_t$)$^7$. As usual, the wage equation depends on a collection of parameters such as the income of an unemployed worker $z_t$, the relative bargaining power of the marginal worker $\nu$, etc... The study of the comparative static and dynamic properties of the economy described by (S1) is carried out in the next two sections under the simplest assumption one can make about the income of an unemployed worker, namely the standard exogenous replacement ratio hypothesis. Under this assumption, $z_t$ merely includes unemployment benefits (in contrast to returns from homework), that are a constant fraction of the wage. Hence we assume in the forthcoming two sections that $z_t = zw_t$, with $0 \le z < 0$ the exogenous replacement ratio. Equation (20) now becomes:

$$ \frac{w_t}{k_t} = \frac{\nu}{1 - z(1 - \nu) - \nu\kappa\theta_t} \cdot f'_2(1, \ell_t). \quad (21) $$

It implies in particular that the wages remains proportional to the state of technical progress, and to the marginal product of labor. This has important consequences that we discuss below. Furthermore, an alternative cost structure is studied in section V.

### III. STEADY STATE AND COMPARATIVE STATICS

When the wage is as given by (21), equation (8) and henceforth system (S1) undergo some transformations. Indeed, if we define for notational brevity $\tilde{\nu} = [1 - z(1 - \nu)]/\nu \ge 1$, (S1) becomes:

$$ \left\{ \begin{aligned} \frac{\dot{\ell}_t}{\theta_t} &= \theta_t q(\theta_t) \cdot (1 - \ell_t) - \mu \ell_t \\ \frac{\dot{\theta}_t}{\theta_t} \cdot \left[ \frac{\kappa \theta_t}{\tilde{\nu} - \kappa \theta_t} - \eta_{\theta,t}^q \right] &= q(\theta_t) \left( \theta_t - \frac{\tilde{\nu} - 1}{\kappa} \right) - \ell_t f'_2(1, \ell_t) \\ &\quad + \chi_t + \mu - \eta_{\ell,t}^{f'_2} \cdot \left( \theta_t q(\theta_t) \cdot \frac{1 - \ell_t}{\ell_t} - \mu \right) \\ \frac{\dot{\chi}_t}{\chi_t} &= \sigma [f'_1(1, \ell_t) - \rho] - f(1, \ell_t) + \chi_t \end{aligned} \right. \quad (S2) $$

with $\eta_{\ell,t}^{f'_2} < 0$ denoting the (not necessarily constant) elasticity of $f'_2$ w.r.t. $\ell$. We now undertake the study of system (S2).

<footnote>$^7$The case of 'monopoly union', $\nu = 1$, where the worker would capture the whole rent from the match, is irrelevant under the assumption of a bargain between the firm and its marginal worker. Moreover, it would lead to an infinite wage and zero vacancy posted.</footnote>
### III.1. Steady state

Steady state values for $\ell_t$, $\theta_t$ and $\chi_t$ – which we shall denote by respectively $\ell^*$, $\theta^*$ and $\chi^*$ – are given by the simultaneous solution of system (S2) above, in which one sets all time derivatives equal to zero. Such a steady state equilibrium characterizes a balanced growth path of the model economy. The first equation of (S2) may be written in a steady state as:

$$ \theta q(\theta) \cdot (1 - \ell) = \mu \ell, \qquad (22) $$

which is well-known as the Beveridge relation. It equalizes the flows in and out of unemployment. The properties of the matching technology $m$ imply that (22) defines a function $\theta = \varphi(\ell)$ which is represented in the $(\ell, \theta)$ plane by an upward sloping, convex schedule that passes through the origin and goes up to infinity as $\ell$ approaches one (see figure 1).

The third equation yields the stationary value of the consumption-capital ratio as a function of employment, $\chi = f(1, \ell) - \sigma[f'_1(1, \ell) - \rho]$. Replacing it and (22) into the second equation, one gets:

$$ q(\theta) = \frac{\kappa}{\tilde{\nu}-1} \cdot \left[ \frac{\mu}{1-\ell} + \sigma\rho + (1-\sigma)f'_{1}(1,\ell) \right], \qquad (23) $$

which again defines a function $\theta = \psi(\ell)$. It is easy to see that $\psi(\ell)$ tends to some finite positive limit as $\ell$ approaches 0, and also that $\psi(\ell)$ goes to zero as $\ell$ gets close to one (see figure 1). Hence the curve that represents function $\psi$ in the $(\ell, \theta)$ plane globally 'looks decreasing', which shows the existence of at least one steady state equilibrium. More precisely, we are able to state the following:

**Proposition 1.** System (S2) has one unique steady state equilibrium, $(\ell^*, \theta^*, \chi^*)$.

**Proof:** first note that $\psi$ is obviously strictly decreasing over the whole $(0, 1)$ interval when $\sigma \le 1$. Since $\varphi$ is increasing, $\varphi(0) < \psi(0)$ and $\varphi(1) > \psi(1)$, the proposition holds true in this case. Now assume $\sigma > 1$. Since $\varphi$ is strictly convex, the strict concavity of $\psi$ is sufficient to show that these functions intersect once and only once. From equation (23) one has:

$$ q' \circ \psi(\ell) \cdot \psi'(\ell) = \frac{\kappa}{\tilde{\nu}-1} \cdot \left[ \frac{\mu}{(1-\ell)^2} - (\sigma-1) f''_{12}(1,\ell) \right]. $$

The right hand side of the latter relation is a strictly increasing function from $(0, 1)$ onto $\mathbb{R}$. Hence once more taking the derivative of this relation yields:

$$ q'' \circ \psi(\ell) \cdot [\psi'(\ell)]^2 + q' \circ \psi(\ell) \cdot \psi''(\ell) > 0, $$
which, together with the negative slope and concavity of $q$, implies that of $\psi$. $\square$

Figure 1 depicts the most likely<fnref n="8" /> case of the steady state equilibrium lying in the decreasing segment of the $\psi$ schedule<fnref n="9" />.

<figure><img src="image_4.png" /><figcaption>Figure 1: steady state equilibrium</figcaption></figure>

Once the steady state values of employment and tightness are known, it is straightforward to derive the value of the consumption-capital ratio and the stationary number of vacancies $v^* = \theta^* (1 - \ell^*)$. The constancy of $\chi$ in the steady state shows that consumption and the capital stock grow at the same rate, $g^* = \sigma [f'_1(1, \ell^*) - \rho]$. This equilibrium growth rate is an increasing function of the steady state level of employment.

### III.2. Comparative statics

The labelled arrows on figure 1 indicate the way in which the $\varphi$ and $\psi$ schedules shift in response to a positive change in the parameters of the model (label $m$ refers to a positive change

<footnote n="8">⁸The proof of this proposition relies heavily on the properties of nonincreasing returns to individual inputs in both production and matching processes. As Mortensen [1989] points out, increasing returns in either technology can give birth to multiple equilibria.</footnote>

<footnote n="9">⁹Something proposition 1 *does not* say is whether or not the steady state point lies in $\psi$'s increasing segment – if such a segment exists, of course. What we argue here is that it would require either a very (that is, unreasonably) high degree of intertemporal substitutability or a very high degree of input complementarity for $\ell^*$ to be in the increasing part of $\psi$. We shall henceforth assume that $\psi'(\ell^*) < 0$, which is the situation depicted on figure 1.</footnote>
in some shift parameter that affects the matching technology). All results are fairly intuitive, and, as far as employment is concerned, unambiguous (see table 1). In particular, the only effect of a rise in the bargaining power of the workers (parameter $\nu$) in this simple model is to raise the bargained wage, henceforth diminishing the demand for labor through a rise in its cost. Unlike Bean and Pissarides [1993], we do not have an ‘anti-Kaldorian’ effect of a rise in the earnings of workers, since we do not assume different saving behaviors for workers and capitalists.

Ambiguity rises, however, when one looks at the response of the equilibrium level of employment to a shift in some neutral productivity parameter, i.e. when one changes $f(\cdot, \cdot)$ into $hf(\cdot, \cdot)$, with $h > 1$. One sees from equation (23) that such a technical shock shifts the $\psi$ schedule upwards (resp. downwards) when $\sigma > 1$ (resp. $\sigma < 1$)$^{10}$. Hence, a positive technical shock may deteriorate the long-run level of employment when intertemporal substitutability is low. The reasons for this are quite clear. A positive productivity shock raises the returns to capital, which has a twofold effect: on the demand side, it encourages savings, which pushes growth upwards and makes the firms more willing to hire. This is some form of a ‘capitalization effect’, according to the endogenous growth literature terminology. On the supply side, it makes capital more expansive and increases the firms’ discount rate, which in turn depresses their demand for labor. The former (positive) effect overshadows the latter (which is negative) whenever the willingness of agents to substitute in time is strong enough.

The response of the growth rate to a positive productivity shift is thus positive when $\sigma \ge 1$. It a priori becomes ambiguous when $\sigma$ is below unity. However, $\theta^*$ decreases in reaction to that shock – as can be seen on figure 1 – as a result of the decline in the firms’ labor demand. As tightness drops, the rate of matching $q(\theta)$ rises. One then sees from equation (23) that the rate of interest finally increases. The positive productivity shock henceforth raises the growth rate, in spite of the fact that it lowers the equilibrium employment rate. This tends to suggest that a negative relationship exists between equilibrium employment and long-run growth, provided that the intertemporal elasticity of substitution is low enough. Note that the origin of the negative relationship between unemployment and growth has nothing to do with labor reallocation problems resulting from the imperfect matching technology. This contrasts with the ‘creative destruction’ effect pointed out in the literature of Schumpeterian inspiration (Aghion and Howitt [1994]). In fact, causality runs from growth to unemployment in those contributions, and reversely in ours.

The response of equilibrium labor market tightness to changes in the set of parameters

<footnote>$^{10}$The same holds true when considering a ‘Harrod-neutral’ shock, changing $f(\cdot, \ell)$ into $f(\cdot, h\ell)$.</footnote>
indicated on the figure is in general of unambiguous sign. The sign pattern of $\theta^*$'s responses is also given in table 1. The signs are unambiguous for shifts in the parameters that affect only the

<table><caption>Table 1: comparative statics</caption><thead><tr><th>change in</th><th>z</th><th>&kappa;</th><th>&nu;</th><th>m</th><th>h</th><th>&sigma;</th><th>&mu;</th><th>&rho;</th></tr></thead><tbody><tr><td>response of l*</td><td>-</td><td>-</td><td>-</td><td>+</td><td>&plusmn; as &sigma; &ge; 1</td><td>+</td><td>-</td><td>-</td></tr><tr><td>response of &theta;*</td><td>-</td><td>-</td><td>-</td><td>+</td><td>&plusmn; as &sigma; &ge; 1</td><td>+</td><td>?</td><td>-</td></tr><tr><td>response of g*</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td><td>-</td><td>-</td></tr></tbody></table>

$\psi$ schedule. The positive response of $\theta^*$ to a positive reallocation shock – a shock on the matching technology – can be shown by eliminating $q(\theta)$ between equations (22) and (23) (see the proof of proposition 2 below). Interpretations of all signs are either straightforward, or discussed at length in a number of earlier contributions among which the formerly cited Pissarides [1990]. For that reason, we shall not go any further in this direction.

## IV. THE DYNAMICS

The study of the dynamics induced by a three-dimensional, forward looking system like (S2), is a practically heavy and complicated, although theoretically standard matter. The aim of this section, however, is to provide some insights into that study. We start with a theoretical analysis of the system's stationary point stability. Some numerical simulations of our model then help to go further into the understanding of its dynamics.

### IV.1. Stability properties of the steady state equilibrium

As we already noted, system (S2) has two forward looking variables (that are $\theta_t$ and $\chi_t$) and one predetermined one ($l_t$). Therefore, what we expect is a local saddle-point stability property. The ensuing proposition exhibits a simple sufficient condition for that:

**Proposition 2.** A sufficient condition for the unique steady state equilibrium ($l^*, \theta^*, \chi^*$) of system (S2) to be locally saddle-point stable is $\sigma \le 1$.

**Proof:** first, we see from equations (22) and (23) that:

$$ \frac{\kappa \theta^*}{\tilde{\nu}} = \frac{l^* (1 - 1/\tilde{\nu})}{1 + [\sigma \rho + (1 - \sigma) f'_1(1, l^*)] \cdot (1 - l^*) / \mu} $$

This expression is below 1 if $\sigma \le 1$. Hence $\kappa \theta^* < \tilde{\nu}$, which is the condition for a positive equilibrium wage, a satisfactory result that will be useful in a while. The second thing
we do is compute the coefficient of the Jacobian matrix of (S2) $M_J(S2)$ in the steady state equilibrium $(\ell^*, \theta^*, \chi^*)$. These coefficients are given in the appendix. Let $P_{S2}(X) = \det\{M_J(S2) - X \cdot I_3\}$ be the characteristic polynomial of the Jacobian matrix of (S2) (with $I_3$ denoting the three dimensional identity matrix). The coefficient values (see appendix) enable us to write the Jacobian of (S2), given by $J(S2) = P_{S2}(0)$. After some calculations, one finds:

$$J(S2) = \frac{(\tilde{\nu} - \kappa\theta^*)\theta^*}{\eta_{\theta}^q\tilde{\nu} - \kappa(1 + \eta_{\theta}^q)\theta^*} \cdot \left\{ \begin{aligned} & [\mu + \theta^*q(\theta^*)] \cdot \left[ q(\theta^*) + \theta^*q'(\theta^*) - \frac{\tilde{\nu}-1}{\kappa} \cdot q'(\theta^*) \right] \\ & + [\theta^*q'(\theta^*) + q(\theta^*)](1-\ell^*)(1-\sigma)f_{12}''(1, \ell^*) \end{aligned} \right\}.$$

All the terms between braces are clearly positive, except possibly $(1 - \sigma)$. It is however positive when $\sigma \le 1$. $J(S2)$ is henceforth negative (remember that $\tilde{\nu} > \kappa\theta^*$ in steady states, as shown above), which implies that $P_{S2}$ has either two complex conjugate roots (with either positive or negative real parts) and one negative real root, or two positive real roots and one negative one, or three negative real roots. To see in which case we are, let us calculate:

$$P_{S2}(\chi^*) = - \frac{(\tilde{\nu} - \kappa\theta^*)\theta^*\chi^*(1-\ell^*)}{\eta_{\theta}^q\tilde{\nu} - \kappa(1 + \eta_{\theta}^q)\theta^*} [q(\theta^*) + \theta^*q'(\theta^*)] [f_2'(1, \ell^*) - \sigma f_{12}''(1, \ell^*)].$$

This has the sign of $f_2'(1, \ell^*) - \sigma f_{12}''(1, \ell^*)$. By the homogeneity of $f$, this is equal to $f_2'(1, \ell^*) + \sigma\ell^*f_{22}''(1, \ell^*)$. This expression is positive for $\sigma \le 1$ since $f$ has decreasing returns w.r.t. labor, implying that $f_2'$'s elasticity w.r.t. $\ell$ is smaller than one in absolute value. Hence, $P_{S2}(0) < 0 < P_{S2}(\chi^*)$, and $P_{S2}$ has one (positive) root between 0 and $\chi^*$. The only possibility for $P_{S2}$ is thus to have one unique negative root and two positive roots, hence the saddle-point property. $\square$

One comment can be made about the latter proposition. The sufficient condition it gives is 'amply' sufficient, which means that the saddle point stability property holds for $\sigma$ in some open neighborhood of unity. Since the most trustable empirical estimations of the intertemporal elasticity of substitution lie 'around or below unity' (Blanchard and Fischer [1989], chapter 2), one may believe that the steady state equilibrium is in fact very likely to exhibit (local) saddle-point stability.

The three dimensions of system (S2) make it difficult to go further into the analysis of its dynamics without using numerical methods. Hence, the next step we take is to simulate the evolution of our economy to provide a more global picture of its dynamic behavior.
IV.2. *Simulations*

To solve numerically our forward-looking system (S2), we use the *time elimination method*.
The basic idea of this method is to transform the boundary value type system formed by (S2),
the initial condition $l_0$ and the transversality conditions associated with the consumers' and the
firms' optimization problems into an initial value type system, which is much more adapted to
numerical solving. The key to this transformation is to rewrite the whole (S2) to obtain a new
system of differential equations in the state variable, $l_t$. This rewriting is equivalent to 'elimi-
nating time' from (S2)<fnref n="11" />. To achieve the numerical solving, we used the ensuing specifications:

• a CES production technology $f(k, \bar{k}l) = A \cdot \left[ \left( (1-\alpha)k \right)^{1-1/\gamma} + (\alpha\bar{k}l)^{1-1/\gamma} \right]^{\gamma/(\gamma-1)}$;

• a Cobb-Douglas matching technology $m(u, v) = u^{\eta} \cdot v^{1-\eta}$ – which implies an isoelastic matching rate $q$ with elasticity $(-\eta)$.

The set of baseline parameter values we adopted for our simulations is summarized in table 2¹². All our results show very satisfactory robustness to changes in the specifications, unless otherwise mentioned<fnref n="13" />. Steady state results for those baseline values are reported in table 3. The unemployment rate is almost 10%, and the growth rate is near 3.6%, a couple of ‘plausible’ values.

<table><caption>Table 2: baseline parameter values</caption>
  <thead>
    <tr>
      <th>Parameter</th>
      <th>Value</th>
      <th>Parameter</th>
      <th>Value</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>α</td>
      <td>0.7</td>
      <td>z</td>
      <td>0.8</td>
    </tr>
    <tr>
      <td>γ</td>
      <td>0.8</td>
      <td>ν</td>
      <td>0.5</td>
    </tr>
    <tr>
      <td>A</td>
      <td>4.5</td>
      <td>μ</td>
      <td>0.065</td>
    </tr>
    <tr>
      <td>η</td>
      <td>0.5</td>
      <td>σ</td>
      <td>1</td>
    </tr>
    <tr>
      <td>κ</td>
      <td>0.5</td>
      <td>ρ</td>
      <td>0.03</td>
    </tr>
  </tbody>
</table>

<footnote n="11">¹¹Since the topic is not central to the paper, we shall not give further details about the method itself. The interested reader should consult Mulligan and Sala I Martin [1993] for a precise description of the method and a number of applications to canonical growth models, or the technical appendix of Barro and Sala I Martin [1995] for a brief presentation.</footnote>

<footnote n="12">¹²Those values come from heterogenous sources. Some were taken from the RBC literature (Fève and Langot [1996], Langot [1995], Langot [1996], Merz [1995]). The elasticity of the aggregate matching function comes close to the empirical findings of Blanchard and Diamond [1989]. Finally, the value of the scale parameter *A* is purely *ad hoc* and meant to imply plausible results for the endogenous variables.</footnote>

<footnote n="13">¹³Changes in all baseline values of table 2 have been tried in a variety of experiments, available on request to the author. Cobb-Douglas with constant returns to scale specification for the aggregate matching technology are supported by the available empirical evidence (Blanchard and Diamond [1989]). Other exercises were also driven with alternative specifications of the matching function (such as one similar to that derived by Pissarides [1979], $m(u, v) = v \cdot (1 - e^{-u/v})$, or a simple CES function). They did not lead to different conclusions.</footnote>
<table><caption>Table 3: steady state results</caption><tr><td>l*</td><td>0.9003</td><td>χ*</td><td>0.0845</td></tr><tr><td>θ*</td><td>0.3443</td><td>g*</td><td>0.0357</td></tr><tr><td>v*</td><td>0.0344</td><td>r*</td><td>0.0657</td></tr></table>

<figure><img src="image_5.png" /><figcaption>Figure 2: the stable arm</figcaption></figure>

The dynamics were simulated by means of the NDSolve subroutine of Mathematica. A collection of graphics is reported in figures 2a through 2d. Figures 2a through 2c are projections of the stable arm in the $(\ell, \theta)$, $(\chi, \theta)$, and $(\ell, \chi)$ planes respectively (with the thick points representing the steady state equilibrium). Figure 2d is the more traditional $(u, v)$ plane representation of the dynamics, with the solid line being the Beveridge curve. Although – merely for the sake of legibility – those figures are restricted to values of $\ell$ within a radius of 5% around $\ell^*$, it turns out that more extreme initial values of $\ell$ also allow the adjustment path to converge to the steady state. For that reason, we may hazard the following:

**Conjecture 3.** Under the condition $\sigma \le 1$, the unique steady state equilibrium $(\ell^*, \theta^*, \chi^*)$ of system (S2) is globally saddle-point stable.
<figure><img src="image_6.png" /><figcaption>Figure 3: the phase diagram – (u, v) plane</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 4: the phase diagram – (ℓ, θ) plane</figcaption></figure>

Once again, the restriction we imposed on the intertemporal elasticity of substitution is sufficient but not necessary. Other experiments with $\sigma$ a little above unity also show global saddle-point stability.

Figure 3 is a projection of the three-dimensional phase diagram of (S2) onto the (u, v) plane. It reports a sample of divergent paths together with the stable arm and the Beveridge curve. Figure 4 is also a projection of the phase diagram onto some (χ = constant) plane. The solid lines on that figure represent the stationaries $\varphi$ and $\psi$ around the steady state. Another striking feature of the simulated dynamics is the quasi-constancy of the tightness parameter along the adjustment path. As figures 2a and 4 show, the slope of this path in the (ℓ, θ) plane is of the order of $10^{-1}$, which may be considered negligible in comparison to the slopes of $\varphi$ and $\psi$ (see figure 4, on which the adjustment path is represented together with the stationaries). Mortensen
[1989] points out – in a model without capital – that the (strict) constancy of $θ_t$ stems from the assumption of constant returns to labor<fnref n="14" />. An interesting result we got here is that (almost) constancy appears to be holding under more general assumptions. It implies, in particular, that such more general assumptions preserve the overshooting feature of vacancies and more specifically the stylized fact of anti-clockwise loops around the Beveridge curve in the $(u, v)$ plane (Hansen [1970]).

When one tries experiments with higher complementarity between capital and labor (i.e. a lower elasticity of substitution $\gamma$), the quasi-constancy observed above turns into more significant positive comovements between $\ell_t$ and $\theta_t$ during their adjustment toward the steady state (see figure 5 for the stable arm obtained in an experiment with $\gamma = 0.15$ and $\kappa$ modified to keep unemployment around 10%; the slope of the adjustment path always remains smaller than that of the $\varphi$ locus, whatsoever). In addition, the adjustment path still 'looks linear'. Henceforth, the terms of 'quasi-linearity' may appear more appropriate to describe the dynamic adjustment than the one of quasi-constancy. The same quasi-linearity property holds true for the consumption-capital ratio, $\chi_t$ (see figures 2b and 2c). This suggests that a linearized version of (S2) should yield a reasonable proxy for the 'true' dynamics of the model<fnref n="15" />.

<figure><img src="image_8.png" /><figcaption>Figure 5: The stable arm with input complementarity</figcaption></figure>

<footnote n="14">¹⁴Indeed, if we assume a constant interest rate and a constant marginal product of labor, $y = f'_2(1, \ell_t)$, it is straightforward to see that, under those assumptions, the dynamics of $\theta_t$ as given by equation (8) and the wage rule (20) are independent and unstable, which implies $\theta_t = \theta^*$ during the adjustment (because the shadow value of a vacant job $\pi_t$ then turns out to be an unstable variable, hence constant along the stable arm, see equation (6)).</footnote>

<footnote n="15">¹⁵Even though the adjustment path reveals itself slightly concave in the $(\ell, \theta)$ plane for values of $\ell$ approaching 0.</footnote>
Moreover, the fact that tightness increases during the adjustment contrasts with the results of Mortensen [1989], who shows that decreasing returns to labor imply negative comovements between $\ell_t$ and $\theta_t$ during adjustment. In fact, the slope of the adjustment path in the $(\ell, \theta)$ plane stems from the combination of a number of conflicting effects. These effects are summarized in the ensuing rewriting of the dynamic equation that governs the evolution of $\theta_t$:

$$ \frac{\dot{\theta}_t}{\theta_t} \cdot \left[ \frac{\kappa \theta_t}{\tilde{\nu} - \kappa \theta_t} - \eta_{\theta,t}^q \right] = \underbrace{f'_1(1, \ell_t) + \mu}_{\text{capital prod. (+)}} - \underbrace{\overbrace{\left[ \frac{\tilde{\nu} - 1}{\kappa} - \theta_t \right] q(\theta_t)}^{\text{tightness (-)}}}_{\text{labor prod. (+)}} - \underbrace{-\eta_{\ell,t}^{f'_2} \cdot \frac{\dot{\ell}_t}{\ell_t}}_{\text{labor prod. (+)}} - \overbrace{\frac{\dot{k}_t}{k_t}}^{\text{cost growth (-)}} \quad (24) $$

First, a 'labor productivity' effect: when employment rises up to its steady state level, the marginal product of labor decreases. This in turn yields a direct negative effect on the demand for labor (since labor is less productive, it is less profitable for firms to post vacant jobs), and an indirect positive 'cost' effect that passes through the wage: the reduction in labor productivity pushes wages (and, from the assumption of proportionality, hiring costs) downwards. Under the assumption of an exogenous replacement ratio, as can be seen in equation (21), wages are proportional to the marginal product of labor. Hence they entirely absorb the variations in labor productivity, thus offsetting the former direct negative effect. This labor productivity effect hence tends to make the slope of the stable arm positive. Second, a 'tightness' effect: the rise in the level of employment, for a given number of vacancies, increases labor market tightness. This tends to increase the average hiring costs ($\kappa_t \theta_t$), thus discouraging the opening of new vacancies. Moreover, higher labor market tightness makes the unemployed better off by increasing their probability of finding a job. This in turn encourages wage pressure and lowers the returns to vacancies. The tightness effect therefore works in the direction of negative comovements between employment and tightness.

The labor productivity and tightness effects are the only ones present in the model without capital that is studied in Mortensen [1989]$^{16}$. It turns out that the negative tightness effect dominates in that model. Now allowing for endogenous capital accumulation and endogenous growth yields two additional effects. A 'cost growth' effect: since the capital stock is a measure of the state of the labor augmenting technical progress, more capital means higher wages and higher hiring costs. When employment goes up, so does the returns to saving, thus encouraging capital accumulation and accelerating cost growth. This tends to discourage the opening of vacancies. Finally, a 'capital productivity' effect: more employment also means higher capital

<footnote>$^{16}$To visualize the model without capital, the reader should just close his eyes and imagine equation (24) with an exogenous interest rate and without the last term.</footnote>
productivity, which in turn fosters hiring. The latter positive effect is obviously the one that dominates in our simulations, and probably in a very general way. Note that the sensitivity of the last two effects mentioned above is increasing in the degree of input complementarity, which explains why the adjustment path becomes steeper and steeper as $\gamma$ decreases.

The evolution of tightness is an important issue, since it drives that of employment. In particular, it has consequences on the degree of unemployment persistence. Indeed, one can derive the speed of adjustment of employment toward its stationary value from equation (9). It's worth:

$$ \left| \frac{\partial \dot{\ell}_t}{\partial \ell_t} \right| = \theta_t q(\theta_t) + \mu. \qquad (25) $$

Hence, an increasing $\theta_t$ means an increasing speed of adjustment. This tends to suggest that unemployment is more and more persistent when it gets bigger, for if $\ell_t$ is way below its steady state value $\ell^*$, then so is $\theta_t$ from $\theta^*$, which implies a reduced speed of adjustment for employment. Another interpretation of this property is that the adjustment of employment are likely to be faster in economic downswings than in booms. This phenomenon is termed steepness asymmetry by Sichel [1993].

The existence of such an asymmetry over the business cycle was already suggested by Keynes in the *General Theory*. It is quite solidly supported by the available empirical evidence (as per instance Neftçi [1984], DeLong and Summers [1986], Sichel [1993], Perraudin [1995]). Moreover, it also appears in theoretical studies of the labor market as a result of the behavior of insiders during the wage setting process (see e.g. Lindbeck and Snower [1989] for a static analysis, and the more recent paper by Huizinga and Schiantarelli [1992] for a dynamic one).

The simulation finally exhibits the usual 'hysteretic' feature of the Romer-style endogenous growth models: the gap between two (otherwise strictly identical) economies' initial capital stocks persists eternally, even if both economies finally grow at the same rate. This is illustrated in figures 6b and 6d. These figures plot the evolution of the capital stock and vacancies<fnref n="17" /> over time for a collection of arbitrarily chosen values of $k_0$. This hysteresis property has another appealing implication in an underemployment context, which is illustrated in figures 6a and 6c. One sees that two economies with identical initial capital stocks, but with different initial unemployment rates end up with different steady state capital stocks and identical steady state

<footnote n="17">¹⁷The plotted variable is in fact the detrended capital stock, $k_t \cdot e^{-g^*t}$.</footnote>
<figure><img src="image_9.png" /><figcaption>Figure 6: hysteresis</figcaption></figure>

unemployment rates. That is, the price to pay for catching up on the labor market is to incur a period of slow capital accumulation, because high unemployment shrinks the pool of savings. In other words, the loss in ‘potentially accumulated’ capital during protracted periods of unemployment is permanent.

V. MORE WAGE STICKINESS

As already noted, an important feature of the latter model is that wages always remain proportional to the productivity of labor, hence absorbing changes in productivity. This feature may be suspected of being responsible for the simplicity of the adjustment dynamics (see e.g. Pissarides [1990], chapter 3), since it clearly appears from a glance at system (S1) that the dynamics of employment are entirely driven by those of vacancies through the labor market tightness parameter, $\theta_t$, which involves the wage/marginal product of labor ratio. Thus, it may be so that more wage stickiness would help making the dynamics of $\theta_t$ (and those of $\ell_t$) a bit more picturesque. Moreover, one can argue on empirical grounds that such wage stickiness would be a desirable feature to endow any dynamic labor market model with. Indeed, it is well known that contrarily to what equation (21) suggests, wages fluctuate by less than the marginal
product of labor. Hence proportionality between $w_t$ and $k_t f'_2(1, \ell_t)$, albeit a necessary long-run property, should be amended in the transitional dynamics. The next subsection discusses the way to introduce such stickiness into system (S1).

## V.1. Leisure as a consumption good

There are several means by which one can obtain such wage rigidity and keep up the long-run indexation of wages on the marginal product of labor. One of them is to drop the assumption of proportionality between hiring costs and wages, and to suppose instead that those costs simply follow the steps of technical change. This assumption is quite frequent, due to its simplicity and the fact that it leaves the steady state of the model, to a wide extent, unchanged. Providing clear theoretical foundations for it nevertheless remains some kind of sport (see Bean and Pissarides [1993]). Another (and maybe more promising) option is to drop the constant replacement ratio hypothesis. Justifications for the non-proportionality of $z_t$ and $w_t$ are not difficult to find. Per instance, Lehmann [1996] argues that unemployment benefits are an endogenous variable that ought to be bargained over at a centralized level. In a totally different order of ideas, Pissarides [1987] and Pissarides [1990] suggest that the employees' threat point in the wage bargaining process reflects more their valuation of time than unemployment benefits. If time (thought of as leisure time) is considered a consumption good, then this 'time valuation' hypothesis suggests proportionality between $z_t$ and the permanent income of an unemployed, $r_t \overline{\Omega}_t^u$. A simple model (that is, with constant returns to labor and no capital) is studied under this assumption by Pissarides [1987] and Pissarides [1990]. We shall now show that the introduction of that kind of wage stickiness in our model is not successful in twisting the dynamics of endogenous variables, hence stressing the robustness of the property of 'quasi-linearity' that appeared in the previous section.

Let $\omega_t = \overline{\Omega}_t^u/k_t$ be the permanent income of an unemployed detrended by the capital stock. Following Pissarides [1987] and Pissarides [1990], we now assume that the unemployed receive no unemployment compensation, and that their instantaneous gain $z_t$ is a fraction of their permanent income, so that $z_t = z \overline{\Omega}_t^u = z\omega_t \cdot k_t$. Replacing this into the wage equation (20), one gets a new expression for $w_t$ involving the detrended permanent income of the unemployed workers:

$$w_t = \frac{z(1-\nu)\omega_t \cdot f'_1(1, \ell_t) + \nu f'_2(1, \ell_t)}{1 - \nu\kappa\theta_t} \cdot k_t. \qquad (26)$$

This expression calls for a couple of remarks. First, one sees that the assumption of zero earnings
for the unemployed stands on the boundary between the exogenous replacement ratio hypothesis (previous section), and the hypothesis we are currently working under. Indeed, setting $z = 0$ in the latter relation takes us back to equation (21). Second, the instantaneous proportionality between the wage and the marginal product of labor is lost in equation (26). Now from (12) and (16), we can easily write the dynamic equation that governs the motion of $\omega_t$:

$$ \frac{\dot{\omega}_t}{\omega_t} = (1-z) \cdot f'_1(1, \ell_t) - \frac{\nu \kappa \theta_t}{1-\nu} \cdot \frac{w_t}{k_t} \cdot \frac{1}{\omega_t} + \chi_t - f(1, \ell_t). \quad (27) $$

With this equation, one sees that the proportionality between $w_t$ and $k_t f'_2(1, \ell_t)$ is restored in a steady state, since the steady state value of $\omega_t$ (say $\omega^*$) given by (27) is clearly proportional to $w_t/k_t$.

With that in mind, we can now write the new dynamic system resulting from the addition of equation (27) to (S1). This addition turns (S1) into a four-dimensional system, which happens to be a little bit more complicated than (S2):

$$ \left\{ \begin{aligned} \dot{\theta}_t &= \theta_t q(\theta_t) \cdot (1 - \ell_t) - \mu \ell_t \\ \frac{\dot{\theta}_t}{\theta_t} \cdot \left[ \frac{\kappa \theta_t}{1/\nu - \kappa \theta_t} - \eta_{\theta,t}^g \right] &= \mu - \ell_t f'_2(1, \ell_t) + \chi_t \\ & \quad + q(\theta_t) \cdot \frac{\nu \kappa \theta_t - \frac{1-\nu}{\kappa} \left[ 1 - z\omega_t \cdot \frac{f'_1(1, \ell_t)}{f'_2(1, \ell_t)} \right]}{1 - (1-\nu) \left[ 1 - z\omega_t \cdot \frac{f'_1(1, \ell_t)}{f'_2(1, \ell_t)} \right]} \\ & \quad - \frac{z(1-\nu) f'_1(1, \ell_t) \cdot \dot{\omega}_t + [z(1-\nu)\omega_t f''_{12}(1, \ell_t) + \nu f''_{22}(1, \ell_t)] \cdot \dot{\ell}_t}{z(1-\nu)\omega_t f'_1(1, \ell_t) + \nu f'_2(1, \ell_t)} \\ \frac{\dot{\chi}_t}{\chi_t} &= \sigma [f'_1(1, \ell_t) - \rho] - f(1, \ell_t) + \chi_t \\ \dot{\omega}_t &= \left( \chi_t - \ell_t f'_2(1, \ell_t) - \frac{z f'_1(1, \ell_t)}{1 - \nu \kappa \theta_t} \right) \cdot \omega_t - \frac{\nu \kappa \theta_t}{1 - \nu \kappa \theta_t} \cdot \frac{\nu}{1-\nu} \cdot f'_2(1, \ell_t) \end{aligned} \right. \quad (S3) $$

We now study this system.

## V.2. Steady state and comparative statics

Again, the steady state of (S3) is characterized by a pair of equations. One is still the Beveridge relation (22). The other stems from a combination of the other three equations of (S3) written with stationary variables. After some tedious algebra, one obtains:

$$ q(\theta) = \frac{\kappa}{1/\nu - 1} \cdot \left[ \frac{\mu}{1-\ell} + \sigma\rho + (1-\sigma) f'_1(1, \ell) + \frac{\mu z \cdot \ell f'_1(1, \ell)}{(1-\ell) \cdot [\sigma\rho - (\sigma + z - 1) f'_1(1, \ell)]} \right], (28) $$

an equation that looks very much like (23), and that defines an implicit function $\theta = \hat{\psi}(\ell)$. Before we completely characterize the steady state equilibrium of (S3), we have to make one
remark: the latter equation consists of (23) plus one additional term (the last ratio in equation (28)). The sign of this extra term's denominator is a priori ambiguous. However, one sees from looking at equation (27) in a steady state that a necessary condition for $w_t$ and $\omega_t$ to have the same (positive) sign in steady states is that this denominator has a positive sign. Hence, to make sure that condition holds true for any possible value of $\ell_t$, we adopt the following:

**Assumption A.** $\sigma + z - 1 > 0$ and $\sigma\rho - (\sigma + z - 1) f'_1(1, 1) > 0$.

The first part of this assumption is not very severe, for we already stressed that a ‘likely’ value of the intertemporal elasticity of substitution should lie around one. The second part just imposes to the returns to capital not to be too strong. Under that assumption, we can state:

**Proposition 4.** Under assumption A, system (S3) has one unique steady state equilibrium, $(\ell^*, \theta^*, \chi^*, \omega^*)$.

**Proof:** the proof runs much like that of proposition 1. From looking at the relative values of $\varphi$ and $\hat{\psi}$ at $\ell = 0$ and $\ell = 1$ respectively, one sees that the concavity of $\hat{\psi}$ is sufficient to prove the proposition. Furthermore, it also results from proposition 1 that a sufficient condition for $\hat{\psi}$ to be strictly concave is that the last ratio in the r.h.s. of (28) is strictly convex. Let $R(\ell)$ be that ratio. We easily see that:

$$R'(\ell) = \frac{\mu z}{(1-\ell) \cdot [\sigma\rho - (\sigma + z - 1) f'_1(1, \ell)]} \times \left\{ \frac{f'_1(1, \ell)}{1-\ell} + \frac{\sigma\rho\ell f''_{12}(1, \ell)}{(1-\ell) \cdot [\sigma\rho - (\sigma + z - 1) f'_1(1, \ell)]} \right\}.$$

From the above assumption, all denominators in the latter expression are clearly decreasing with respect to $\ell$. $f'_1$ is increasing and, from the assumption of decreasing returns to individual inputs, so is $\ell f''_{12}$. $R(\ell)$ is thus strictly convex. $\square$

The comparative static properties of this new model economy are much like those of the former (see section III.2 above). The mere difference is that the threat point of workers in the wage bargaining process is now directly influenced by the rate of returns to savings: a higher rate of interest means a higher permanent income for the unemployed, hence higher wages. This adds a negative influence over the long-run employment level of the neutral productivity parameter. Like in the former model, a positive shift in that parameter fosters the returns to capital, and this now has three effects: one encourages hiring through a kind of ‘capitalization effect’, the second discourages hiring by making the use of capital more expansive, and the third one is the
one just mentioned, and is also negative. This makes the possibility of a negative relationship
between long-run employment and growth that we stressed two sections ago even more likely.

V.3. *The dynamics*

Before showing the results of some numerical experiments, we can have a quick discussion
about the stability properties of the unique steady state equilibrium of (S3). Formally, we can
state the following:

**Proposition 5.** Under assumption A, and if the unique steady state equilibrium of system (S2) is saddle-point stable, then the unique steady state equilibrium of (S3) is saddle-point stable for a sufficiently small valuation of leisure, (small z). In other words, there exists $\bar{z} > 0$ such that $z < \bar{z}$ implies the saddle-point stability property for $(\ell^*, \theta^*, \chi^*, \omega^*)$.

**Proof:** in the $z=0$ case, we saw that the first three equations of (S3) were the same as those of (S2). Furthermore in that case, the last equation in (S3) indicates that $\omega_t$ has unstable dynamics. Hence, if $z=0$ and if the stationary point of the corresponding (S2)-like system is a saddle, then the Jacobian matrix of (S3) at its unique steady state equilibrium has exactly three positive eigenvalues, and one negative one. For all coefficients of that matrix are continuous functions of $z$, so are its eigenvalues. Hence they keep their signs in some open neighborhood of $z=0$. $\square$

In fact, the saddle point property of (S3) is very likely. Our guess is that it holds under very general assumptions, although we do not provide a complete rigorous proof<fnref n="18" />.

We now turn to simulations. The specifications adopted for this new set of exercises are basically the same as those of the previous section. We just change the value of $z$ to $z = 0.1$, and that of $\nu$ to $\nu = 0.8$ (hence the 'strong worker' label). The steady state results are reported in table 4. A projection of the corresponding phase diagram onto some ($\chi$ = constant) plane is given in figure 7 (the solid lines again represent the stationary schedules $\varphi$ and $\hat{\psi}$. As in the case of an exogenous replacement ratio, the system seems to show global saddle-point stability. Furthermore, the stable arm is still slightly increasing and still looks 'quasi-linear'. The remarkable robustness of this feature may again be used as an argument in favor of the study of linearized dynamics.

<footnote n="18">¹⁸It is possible to show quite easily that the jacobian of (S3) is unambiguously negative under assumption A, which implies that the jcaobian matrix has one negative and one positive eigenvalue. We conjecture that the remaining two eigenvalues are both real and positive. We have no proof for that, however, since the calculations it would take to provide such a proof are far away beyond our capacities.</footnote>
<table><caption>Table 4: steady state results – strong worker</caption><tr><td>l*</td><td>0.9006</td><td>χ*</td><td>0.0844</td></tr><tr><td>θ*</td><td>0.3468</td><td>g*</td><td>0.0357</td></tr><tr><td>v*</td><td>0.0345</td><td>ω*</td><td>1.7443</td></tr></table>

<figure><img src="image_10.png" /><figcaption>Figure 7: the phase diagram – (l, θ) plane, strong worker</figcaption></figure>

Things change a little, however, in the case of a smaller weight of the worker in the bargaining process. Figure 8 plots the stable arm for the value $v = .1$ (and $κ$ adapted to recover the unemployment rate of 10%). One sees that it becomes slightly decreasing in the $(l, θ)$ plane. What happens is that dropping the assumption of proportionality between wages and labor productivity adds a negative labor productivity effect to the dynamics of $θ_t$ that was entirely offset in the former model (see previous section). Since the non-proportionality comes from the specification of the worker's threat point, it becomes stronger when the worker has less weight in the bargain, since his earnings then get closer to his threat point (see equation (26)).

## VI. CONCLUDING REMARKS

This article has provided a theoretical study of the ‘matching’ model of equilibrium unemployment with endogenous capital accumulation and endogenous growth à la Romer [1986]. It has shown, under general assumptions, the existence and uniqueness of a balanced growth path with a stationary rate of unemployment. The comparative static properties of this balanced growth path suggest that a negative correlation between the long-run rate of growth and the long-run level of employment may be observed. Concerning the dynamics, it was shown that the balanced growth path was (in general) saddle-point stable. Finally, a collection of numerical
<figure><img src="image_11.png" /><figcaption>Figure 8: the stable arm – (ℓ, θ) plane, weaker worker</figcaption></figure>

experiments uncovered appealing dynamic features of the model, such as the apparently very robust ‘linear look’ of its stable arm. This suggests that linearizing the model – as is usually done for instance by practitioners of the RBC methodology – should yield a good proxy of its ‘true’ dynamics. The model also exhibits some dynamic asymmetry, a feature that is supported by considerable empirical evidence.

The theoretical models of the links between growth and employment is an area where much work remains to be done. In particular, a nice (although probably tricky) issue is the dynamic study of models with ‘backward’ causality, i.e. running from growth to unemployment. This can be done by endogenizing the rate of structural change, as in Mortensen and Pissarides [1994] and Mortensen and Pissarides [1995], or by studying phenomena of ‘creative destruction’ along the lines drawn by Aghion and Howitt [1994].

Finally, as noted by Bean and Pissarides [1993], this model is strongly Classical, since a fall in the propensity to save (e.g. through a fall in the intertemporal elasticity of substitution) unambiguously reduces the growth rate. These authors show (in a steady state analysis) that introducing Keynesian demand effects in the model by positing an imperfectly competitive goods market yields a possibly positive correlation between the propensity to consume and the long-run rate of growth. This positive correlation seems to be supported by empirical evidence (Caballero [1993]). Such Keynesian effects might have interesting dynamic effects, since the propensity to consume actually increases during the dynamic adjustment of the variables in our model.
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# APPENDIX

## A. THE JACOBIAN MATRICES

To avoid heaviness, we will omit the star (*) superscripts, although all values are taken in the steady state. The coefficients of the Jacobian matrix of (S2), $M_J(S2)$, are as follows:

$$ \frac{\partial \dot{l}}{\partial l} = -\theta q(\theta) - \mu < 0, $$

$$ \frac{\partial \dot{l}}{\partial \theta} = [q(\theta) + \theta q'(\theta)] \cdot (1 - l) > 0, $$

$$ \frac{\partial \dot{l}}{\partial \chi} = 0, $$

$$ \frac{\partial \dot{\theta}}{\partial l} = \frac{(\tilde{\nu} - \kappa\theta) \cdot \theta}{\eta_{\theta}^q \tilde{\nu} - \kappa(1 + \eta_{\theta}^q)\theta} \cdot \left[ -\frac{f_{22}''(1, \ell)}{f_2'(1, \ell)} \cdot [\mu + \theta q(\theta)] + f_2'(1, \ell)(1 + \eta_{\ell}^{f_2'}) \right] < 0, $$

$$ \frac{\partial \dot{\theta}}{\partial \theta} = \frac{(\tilde{\nu} - \kappa\theta) \cdot \theta}{\eta_{\theta}^q \tilde{\nu} - \kappa(1 + \eta_{\theta}^q)\theta} \cdot \left[ \frac{\tilde{\nu} - 1}{\kappa} \cdot q'(\theta) + [\theta q'(\theta) + q(\theta)] \cdot \left\{ \frac{1-\ell}{\ell} \cdot \eta_{\ell}^{f_2'} - 1 \right\} \right] \le 0, $$

$$ \frac{\partial \dot{\theta}}{\partial \chi} = - \frac{(\tilde{\nu} - \kappa\theta) \cdot \theta}{\eta_{\theta}^q \tilde{\nu} - \kappa(1 + \eta_{\theta}^q)\theta} > 0, $$

$$ \frac{\partial \dot{\chi}}{\partial l} = [\sigma f_{12}''(1, \ell) - f_2'(1, \ell)] \cdot \chi \le 0, $$

$$ \frac{\partial \dot{\chi}}{\partial \theta} = 0, $$

$$ \frac{\partial \dot{\chi}}{\partial \chi} = \chi > 0. $$

The Jacobian matrix of system (S3), $M_J(S3)$ is obtained by adding or replacing the relevant coefficients by the ones that follow:

$$ \frac{\partial \dot{l}}{\partial \omega} = 0, $$

$$ \frac{\partial \dot{\theta}}{\partial l} \times A \times B = -B[f_1'(1, \ell) + \ell f_{22}''(1, \ell)] + \frac{q(\theta)}{\kappa B} z(1 - \nu) \omega (1 - \nu \kappa \theta) \cdot \frac{f_{12}'' \cdot f_2' - f_1' \cdot f_{22}''}{(f_2')^2} \\ -z(1 - \nu) \frac{f_1'(1, \ell)}{f_2'(1, \ell)} \cdot \frac{\partial \dot{\omega}}{\partial l} - \left[ z(1 - \nu) \omega \frac{f_{12}''(1, \ell)}{f_2'(1, \ell)} + \nu \frac{f_{22}''(1, \ell)}{f_2'(1, \ell)} \right] \cdot \frac{\partial \dot{l}}{\partial l} \le 0, $$

$$ \frac{\partial \dot{\theta}}{\partial \theta} \times A \times B = \frac{1}{\kappa} \cdot [q'(\theta) \cdot (\nu\kappa\theta - 1 + B) + \nu\kappa q(\theta)] - z(1 - \nu) \frac{f_1'(1, \ell)}{f_2'(1, \ell)} \cdot \frac{\partial \dot{\omega}}{\partial \theta} \\ - \left[ z(1 - \nu) \omega \frac{f_{12}''(1, \ell)}{f_2'(1, \ell)} + \nu \frac{f_{22}''(1, \ell)}{f_2'(1, \ell)} \right] \cdot \frac{\partial \dot{l}}{\partial \theta} \le 0, $$
$$ \frac{\partial \dot{\theta}}{\partial \theta} \times A \times B = \nu > 0, $$

$$ \frac{\partial \dot{\theta}}{\partial \omega} \times A \times B = \left[ \frac{q(\theta)}{\kappa B} (1 - \nu\kappa\theta) + \frac{\partial \dot{\omega}}{\partial \omega} \right] \cdot z(1 - \nu) \frac{f_1'(1, \ell)}{f_2'(1, \ell)} > 0, $$

$$ \frac{\partial \dot{\chi}}{\partial \omega} = 0, $$

$$ \frac{\partial \dot{\omega}}{\partial \ell} = \left[ 1 - \frac{z}{1 - \nu\kappa\theta} \right] \omega f_{12}''(1, \ell) - \frac{\nu}{1 - \nu} \cdot \frac{\nu\kappa}{1 - \nu\kappa\theta} \cdot f_{22}''(1, \ell) - \omega f_{2}'(1, \ell) \le 0, $$

$$ \frac{\partial \dot{\omega}}{\partial \theta} = - \frac{\nu\kappa}{(1 - \nu\kappa\theta)^2} \cdot \left[ z\omega f_1'(1, \ell) + \frac{\nu}{1 - \nu} \cdot f_2'(1, \ell) \right] < 0, $$

$$ \frac{\partial \dot{\omega}}{\partial \chi} = \omega > 0, $$

$$ \frac{\partial \dot{\omega}}{\partial \omega} = \left[ 1 - \frac{z}{1 - \nu\kappa\theta} \right] f_1'(1, \ell) + \chi - f(1, \ell) = \frac{\nu}{1 - \nu} \cdot \frac{\nu\kappa}{1 - \nu\kappa\theta} \cdot f_2'(1, \ell) > 0, $$

with:

$$ A = \frac{\eta_{\theta}^{q}/\nu - \kappa(1 + \eta_{\theta}^{q})\theta}{(1/\nu - \kappa\theta)\cdot\theta} < 0, \quad \text{and} \quad B = \nu + z(1 - \nu)\omega \frac{f_{1}'(1, \ell)}{f_{2}'(1, \ell)} > 0. $$