# Unemployment, Education and Growth

Fabien Postel-Vinay

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CAHIERS ECO & MATHS

97.26

Unemployment, Education and Growth

Fabien POSTEL-VINAY

MAD
UNEMPLOYMENT, EDUCATION AND GROWTH

Fabien Postel-Vinay

MAD, Université de Paris I*†

January 1997

ABSTRACT

This paper incorporates endogenous education in a standard 'matching and bargaining' unemployment model. It shows that, despite the positive influence of the individual schooling effort on individual job opportunities, the average duration of studies in the economy does not affect the equilibrium matching rate of individual workers. It does (ambiguously) affect the labor market, however, by making labor more productive and at the same time more expensive. This ambiguity may give birth to multiple steady state equilibria of the model. We finally turn to growth issues, and show the existence of a positive feedback between employment, education and growth.

**Key words:** Schooling, human capital, unemployment, matching, growth.

**JEL classification:** E24, J41, I20.

RÉSUMÉ

Cet article incorpore des choix endogènes d'éducation dans le modèle “d'appariement-négociations”. Il est montré que, bien que l'effort individuel d'éducation augmente la probabilité individuelle d'embauche, la durée moyenne des études n'affecte pas le taux d'embauche moyen de l'économie. Elle influence cependant le chômage d'équilibre de façon ambiguë en rendant le facteur travail à la fois plus productif et plus onéreux. Cette ambiguïté donne naissance à des équilibres multiples. Ces résultats se retrouvent dans une économie en croissance, où nous montrons l'existence d'un effet de retour positif entre l'éducation, la croissance et l'emploi.

**Mots-clefs :** Education, capital humain, chômage, appariement, croissance.

**Classification JEL :** E24, J41, I20.

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

<footnote>†Beneficial comments from A. d'Autume, P. Cahuc, M. Carré, H. Kempf, E. Lehmann and M. Pucci on earlier drafts of this paper are gratefully acknowledged. Of course I'm the only one to blame for any remaining error.</footnote>
# 1. INTRODUCTION

Studying the connections between the educational sector and the labor market is a major political and ‘practical’ economic issue. Policies to encourage education, as an attempt to cut unemployment, have repeatedly been undertaken in most European countries since the late seventies. The idea underlying those attempts is probably that there is a (positive) feedback between technological progress, labor productivity and the increase in skill requirements. Despite its obvious practical importance, not many theoretical approaches to that matter have been undertaken, so far.

Important contributions nevertheless exist, which can roughly be arranged in two classes. The first class (Cahuc and Michel [1996], Saint-Paul [1994]) is related to the unskilled unemployment literature and simply adds endogenous educational choices to it. It then studies the effects of the incentives to become skilled on the unemployment rates of skilled and unskilled workers and on the skill structure of the labor force. The second class (Laing, Palivos and Wang [1995]) merely considers one type of labor, and focuses on the returns to education. The latter paper incorporates human capital accumulation in a matching model of unemployment à la Diamond [1982] and Pissarides [1990]. Basically, what it shows is that a costly schooling effort entails better earnings once employed, as a consequence of the rise in productivity investment in human capital entails. Since this is the only incentive to provide a schooling effort it considers, that model ignores the direct ‘allocative’ effect on job opportunities of a higher investment in human capital. It is indeed a well-known fact that education enhances individual chances of finding a job, as well as earnings<fnref n="1" />. The present paper is an attempt to recover this effect.

We show that investment in human capital is beneficial to workers in two respects: first, it gives them better job opportunities by directly raising their probability of being matched with a firm, and in second place, it raises their expected wage. The former effect reflects the fact that agents know they wont be profitable to any firm if they do not acquire enough human capital. Thus, although in equilibrium the average level of schooling — as measured by the average length of schooling time — has influence neither on the aggregate matching rate, nor on the so-called ‘Beveridge curve’, individuals do not take their own probability of being hired as given.

The framework we find adapted to uncover that effect is the ‘stochastic matching’ version

<footnote n="1">¹Education also provides social status, that can be thought of as raising the agents’ welfare (see e.g. Fershtman, Murphy and Weiss [1996] for theoretical insights on that topic). We shall ignore this particular point.</footnote>
of the basic matching model (Pissarides [1990], chapter 5). Although its writing may appear
temporarily heavy, the model gives very simple and intuitive equilibrium results that come close
to those of Laing et al. [1995] — with slight differences in their economic meanings, though. In
particular, the model shows the possible coexistence of multiple steady state equilibria. Those
arise because education influences the labor market in an ambiguous way: it makes labor more
productive one the one hand, and more expensive on the other. We suggest the possibility for
a government to keep the economy on the ‘rich’ equilibrium, i.e. the one with a high level
of activity on the labor market and a low unemployment rate. That can be done either by
improving the search market efficacy, or simply by putting legal constraints on a minimum
schooling duration.

The final concern of this paper is the rate of growth. It is made endogenous<fnref n="2" /> by considering
the external effects associated with education as Uzawa [1965] and Lucas [1988] primarily sug-
gested. We show that a higher employment level, by encouraging education, also fosters growth.
Growth makes agents more patient, which leads them to undertake longer studies. Hence, there
is a positive feedback between employment, education and growth.

The paper is organized as follows: the next section describes the basic structure of the
model. Section 3 studies equilibria with an exogenous investment in human capital. Section 4
endogenizes the level of schooling effort, and shows the existence of multiple equilibria. Section
5 handles issues of growth. Section 6 concludes.

2. THE BASIC FRAMEWORK

Time is continuous, and the relevant variables are subscripted by time, *t*. The economy is comprised of an educational sector and a productive sector. In the productive sector, there is an endogenously sized continuum of identical small firms indexed *j* that possess a fixed-coefficient production technology requiring a single worker. Hiring a worker implies opening a vacant job slot and engaging into job advertising. The cost per unit time of job advertising is assumed constant and designated by *κ* > 0. We assume that, upon paying *κ*, any number of firms can open a job vacancy and search for unemployed workers.

The demographic structure of the model borrows from the ‘perpetual youth’ model by Blanchard and Fischer [1989], chapter 2. The economy is populated by a continuum of individuals

<footnote n="2">²This is another respect in which our model departs from that of Laing *et al.* [1995]. Although these authors make *individual* productivities grow at an endogenous rate, the *aggregate* growth rate is exogenous in their model (and equals the demographic growth rate). This simply results from the absence of any sort of capital good in their model.</footnote>
facing a positive probability flow of dying at any moment. This probability — say, $n$ — is assumed constant throughout life. To keep a constant population (the size of which we shall normalize to one), we assume that $n$ is also the rate at which individuals are being born. Individuals are a priori identical. When being born, any agent<fnref n="3" /> $i$ immediately enters the educational sector, where she/he stays for a period of length $h_i$. Time spent at school is transformed into human capital by the educational system, so that Mrs./Mr. $i$ comes out of it<fnref n="4" /> with a diploma worth $h_i$ units of labor. Hence, workers may differ after their schooling period by the levels of human capital they have acquired.

Once agents have completed their schooling, they enter the labor market as unemployed workers and search for a vacant job to be matched with. The background of the matching process we posit is the ‘stochastic job matching’ model of Pissarides [1990], chapter 5. Suppose there are $\bar{u}_t$ unemployed workers and $\bar{v}_t$ vacant jobs in the economy<fnref n="5" /> at time $t$. We assume that firms and unemployed workers meet each other at a rate $\mu(\bar{u}_t, \bar{v}_t)$, that is, there are $\mu(\bar{u}_t, \bar{v}_t)$ firm-worker contacts per unit time. About function $\mu$, we make the following usual restrictions: $\mu$ is $C^\infty$, strictly increasing and strictly concave, and exhibits constant returns to scale. It also satisfies the boundary conditions $\mu(0, \bar{v}_t) = \mu(\bar{u}_t, 0) = 0$, $\lim_{x\to 0} \partial\mu/\partial x = +\infty$, $\lim_{x\to +\infty} \partial\mu/\partial x = 0$ for $x = \bar{u}_t, \bar{v}_t$.

Once a firm and a worker have met, they must decide whether to accept or to reject the match. Following Pissarides [1990], chapter 5, we assume that job-worker pairs differ in their productivity. The productivity of agent $i$, when working in firm $j$, is assumed to be worth $\varepsilon_{ij}h_i$. Parameter $\varepsilon_{ij}$ is random and specific to the job-worker pair. It is revealed both to the firm and to the worker once they have met each other. It is distributed over $\mathbb{R}_+$ according to some known-by-all stationary function $\Phi' = \varphi$ that does not differ across matches, and the support of which we assume unbounded from above. Hence, not all matches are acceptable to firms and workers. Low-productivity matches may not compensate either one of the partners for the costs incurred in forming the match. More precisely, the value at time $t$ to firm $j$ of a match with worker $i$, $\Pi_{ij,t}^o$, obeys the following asset equation:

$$r\Pi_{ij,t}^o = \varepsilon_{ij}h_i - w_{ij,t} + n(\Pi_t^v - \Pi_{ij,t}^o) + \dot{\Pi}_{ij,t}^o, \quad (1)$$

where $r$ denotes the (constant and exogenous) rate of discount, that is common to all agents in

<footnote n="3">³There are $n$ agents being born at any date $\tau$. Hence, $i \in [0, n]$ should clearly be subscripted by $\tau$, as it denotes the $i$th agent of the $\tau$-generation. To avoid heaviness, we will keep the information about agent $i$'s generation — that is, the subscripting of $i$ by $\tau$ — implicit. It is clear that this omission will entail no loss of information in steady states, since all agents are identical and are then being born in identical conditions.</footnote>

<footnote n="4">⁴Provided she/he did not die during that time, an event happening with probability $1 - e^{-nh_i}$.</footnote>

<footnote n="5">⁵Bars denote aggregate variables, taken as given by individuals.</footnote>
the economy, $w_{ij,t}$ is the wage paid at date $t$ by firm $j$ to worker $i$, and $\Pi_t^v$ is the asset value of a vacant job slot (which does not depend on the firm index $j$, since it was assumed that all firms were alike). Note that $n$ appears as an exogenous rate of job termination$^6$. The asset equation for $\Pi_t^v$ writes down as:

$$r\Pi_t^v = -\kappa + q_t \cdot (\top \Pi_t^o - \Pi_t^v) + \dot{\Pi}_t^v, \quad (2)$$

with $q_t$ being the rate at which a vacant job is successfully matched with a worker, and $\top \Pi_t^o = E_{h,\epsilon}(\Pi_{ij,t}^o | \Pi_{ij,t}^o \ge \Pi_t^v)$ being the expected value of an occupied job (which again does not depend on the particular firm). Note that the schooling effort of the incoming worker is a priori random from the firm's point of view, so that the above expectation has to be taken with respect to the random variable $(h, \epsilon)$. Since there is free entry on the search market, the equilibrium value of $\Pi_t^v$ is zero. Furthermore, by accepting a particular match, firm $j$ wastes $\Pi_t^v$ and receives $\Pi_{ij,t}^o$. It will thus accept any match that yields positive profits.

We now turn to the worker. From her/his match with firm $j$, Mrs./Mr. $i$ will get a level of utility that obeys$^7$:

$$(r + n) \Omega_{ij,t}^e = w_{ij,t} - b_t + \dot{\Omega}_{ij,t}^e, \quad (3)$$

where $b_t$ is a lump-sum tax meant to finance the unemployment compensation system. Note that it is never in the interest of a firm to fire its worker before she/he dies, for it would then have to re-pay for the opening of a vacant job and re-enter the search market, which would yield negative profits. Let $\Omega_{i,t}^u$ be the asset value for worker $i$ of being unemployed. (Note that the latter value does depend on $i$, since workers may differ in their abilities, $h_i$, which implies that they may not have equal job opportunities when unemployed). $\Omega_{i,t}^u$ satisfies:

$$(r + n) \Omega_{i,t}^u = \bar{z}_t - b_t + p_{i,t} \cdot (\top \Omega_{i,t}^e - \Omega_{i,t}^u) + \dot{\Omega}_{i,t}^u, \quad (4)$$

with $p_{i,t}$ denoting the instantaneous probability for worker $i$ to be matched, $\bar{z}_t$ the amount of per unit time unemployment compensation, and $\top \Omega_{i,t}^e = E_\epsilon (\Omega_{ij,t}^e | \Omega_{ij,t}^e \ge \Omega_{i,t}^u)$ the expected value of being employed for this worker (which also depends on $i$). Of course, all individuals have knowledge of their levels of human capital a priori. Hence, they take their expectations with respect to the sole random variable $\epsilon$. Now, it is clear that worker $i$ is willing to accept any wage $w_{ij,t}$ such that $\Omega_{ij,t}^e \ge \Omega_{i,t}^u$.

<footnote>$^6$Assuming infinitely-lived workers and an exogenous rate of ‘structural change’ at which jobs are destroyed would have been perfectly equivalent to our ‘perpetual youth’ hypothesis.</footnote>

<footnote>$^7$For the sake of simplicity, workers are assumed risk-neutral.</footnote>
We now restrain the scope of the analysis to steady states of the model. Since all variables are now held constant over time, we may drop time subscripts. The constancy of wages implies that the discounted sum of profits accruing to firm *j* from its match with worker *i* is proportional to $\varepsilon_{ij}h_i - w_{ij}$ (see equation (1)). That firm will thus take any worker *i* such that $\varepsilon_{ij}h_i \ge w_{ij}$. Moreover, it is now possible to define the reservation wage of any worker *i* — say, $w_i^r$ — as the wage that equalizes $\Omega_i^u$ and the steady state asset value of being matched with a firm that pays $w_i^r$. From (3) and (4), one sees that it is defined by:

$$w_i^r = \bar{z} + p_i \cdot (\perp \Omega_i^e - \Omega_i^u). \qquad (5)$$

It is clear that the reservation wage of worker *i* increases w.r.t. the probability of this worker to be properly matched, as well as to her/his expected earnings. It also depends positively on the amount of unemployment benefits.

In a steady state, worker *i* accepts the match with firm *j* for any (stationary) wage profile such that $w_{ij} \ge w_i^r$, whereas the firm takes it for any wage profile such that $w_{ij} \le \varepsilon_{ij}h_i$. Hence the (*i*, *j*) worker-firm pair will form a new match if and only if $w_i^r \le \varepsilon_{ij}h_i$. That is, a match will actually occur every time that a firm meets a worker with a reservation wage that does not exceed the productivity of the potential match. This implies that the firm and the worker both know whether or not they should accept the match a priori (i.e. before they start thinking of what the wage might be). Moreover, they always agree on that point.

Once the firm and the worker have agreed upon working with each other, they start bargaining over the wage the worker will receive. As is traditional, we shall assume that the wage is the outcome of a generalized Nash bargain that splits the surplus accruing from the match into two shares such that:

$$\frac{\Omega_{ij}^{e} - \Omega_{i}^{u}}{\nu} = \frac{\Pi_{ij}^{o} - \Pi_{i}^{v}}{1 - \nu}. \qquad (6)$$

(Note that the same equality applies out of steady states.) Taking the ($\perp$)-conditional mean of the bargain outcome (6) yields $(1 - \nu) \cdot (\perp \Omega_i^e - \Omega_i^u) = \nu \cdot \perp \Pi_i^o$. Substituting equations (1) through (4) together with the free entry condition $\Pi^v = 0$ in the latter relation, one gets, after some algebra:

$$\perp w_i = \frac{(r + n + p_i) \cdot \nu^{\perp} \varepsilon_i h_i + (1 - \nu)(r + n) \bar{z}}{r + n + \nu p_i}, \qquad (7)$$

the expression of agent *i*'s expected wage. Now the ($\perp$)-conditional mean of (6) also implies

$$\perp w_i = \nu^{\perp} \varepsilon_i h_i + (1 - \nu) [\bar{z} + p_i \cdot (\perp \Omega_i^e - \Omega_i^u)] = \nu^{\perp} \varepsilon_i h_i + (1 - \nu) w_i^r. \qquad (8)$$
(Note that the latter relationship remains valid out of steady states.) This, together with (7),
implies that the reservation wage of worker *i* is given by:

$$
w_i^r = \frac{(r+n)\bar{z} + \nu p_i \cdot \perp \varepsilon_i h_i}{r+n+\nu p_i}. \qquad (9)
$$

Remembering the definition of the bottom (⊥) operator, it is straightforward to see that it is equivalent to the expectation operator, taken conditionally on the acceptance of the match by both parties, $E_ε(\cdot|\varepsilon_{ij} \ge w_i^r/h_i)$. Hence, $\perp \varepsilon_i = E_ε(\varepsilon_{ij}|\varepsilon_{ij} \ge w_i^r/h_i)$ is clearly a function of $w_i^r/h_i$, which characterizes (9) as a first relationship between that variable and $p_i$.

Now, worker *i* is actually hired when she/he meets a firm, which happens with an instan-
taneous probability μ(1, v̄/ū), provided that her/his reservation wage does not exceed her/his
productivity, an event that has an instantaneous probability of 1 − Φ(ŵᵢ²/hi) in a steady state.
From that, we can derive the steady state rate at which any unemployed worker *i* is matched:

$$
p_i = \mu \left(1, \frac{\bar{v}}{\bar{u}}\right) \cdot \left[1 - \Phi\left(\frac{w_i^r}{h_i}\right)\right]. \qquad (10)
$$

Equation (10) defines the individual probability of being hired as a decreasing function of $w_i^r/h_i$.
This conducts us to state the following:

**Proposition 1.** Equations (9) and (10) uniquely define a steady state reservation wage $w_i^r$ and a steady state individual matching rate $p_i$ as derivable functions of $\bar{z}$, $h_i$, $r$, $n$, $\nu$, and $\bar{v}/\bar{u}$. The partial derivatives have the following signs:

$$
\frac{\partial w_i^r}{\partial h_i}, \frac{\partial w_i^r}{\partial \bar{z}}, \frac{\partial w_i^r}{\partial \nu}, \frac{\partial w_i^r}{\partial (\bar{v}/\bar{u})} \text{ and } \frac{\partial p_i}{\partial h_i}, \frac{\partial p_i}{\partial r}, \frac{\partial p_i}{\partial n} \text{ positive;}
$$

$$
\frac{\partial w_i^r}{\partial r}, \frac{\partial w_i^r}{\partial n} \text{ and } \frac{\partial p_i}{\partial \bar{z}}, \frac{\partial p_i}{\partial \nu} \text{ negative.}
$$

Proof: Please consult the appendix.

<figure><img src="image_2.png" /></figure>

As proposition 1 shows, the reservation wage $w_i^r$ depends positively on $h_i$. This reflects the fact that schooling is beneficial to workers in two respects: it increases their individual probability of finding a job, and, by raising their productivity, it also makes their average earnings higher. With knowledge of that positive dependence, agent *i* will be able to maximize her/his discounted present wealth at the date of her/his birth by setting the marginal cost of her/his period spent at school equal to its marginal reward. Before we address this problem, it is useful to study the equilibria of our economy under the simple assumption of an exogenous level of schooling effort.
### 3. STEADY STATE EQUILIBRIUM WITH AN EXOGENOUS SCHOOLING EFFORT

We now focus on symmetric steady state equilibria. Since all workers are identical, they all choose the same duration for their studies — say $h$ — in such an equilibrium. They also have equal probabilities of matching, receive the same wage on average and accept wages above the same reservation value, which allows us to drop all subscripts in equilibrium. This section is devoted to the study of equilibria with an exogenously given $h$. The endogenous determination of the schooling effort is carried out in the next section.

Considering symmetric equilibria nicely simplifies the analysis. As we saw in the previous section, out of such an equilibrium, workers are heterogenous in two respects from the representative firm's point of view. They differ in the productivity of the virtual matches the firm could form with them, and they differ in their levels of human capital, $h_i$. Before the representative firm and some particular individual $i$ meet, $h_i$ is a random variable for the former, whereas it is known to the latter. In other words, even if the firm can observe with certainty every individual schooling effort, it has knowledge neither of which particular worker it will next meet, nor of what the human capital of this worker will be. In a symmetric equilibrium, though, the distribution of the schooling effort becomes trivial, and the asymmetry vanishes. The productivity parameter $\varepsilon_{ij}$ is the only source of heterogeneity left, and it is 'equally random' for the firm and for the worker. It then becomes true that, according to the bargain outcome (6), the top ($\top$) operator that we used for firms and the bottom ($\bot$) operator that we used for workers are both equivalent to $E_\varepsilon (\cdot | \varepsilon_{ij} \ge w^r/h)$. We shall denote this operator by a tilde, thus writing $\tilde{x} = E_\varepsilon (x | \varepsilon_{ij} \ge w^r/h)$ for any random variable $x$.

An additional assumption we make is that the amount of unemployment benefits, $\bar{z}$, is proportional to the average expected wage<fnref n="8" />. Denoting by $z \in [0, 1)$ the constant replacement ratio, one has $\bar{z} = z \cdot \tilde{w}$. Taking account of that hypothesis in (7) gives the equilibrium expected wage:

$$ \tilde{w} = \frac{r+n+p}{\nu p + (r+n)(1-z+z\nu)} \cdot \nu \tilde{\varepsilon} \cdot h \quad (11) $$

which, together with (9) yields the equilibrium value of the reservation wage:

$$ w^r = \frac{p+z(r+n)}{\nu p + (r+n)(1-z+z\nu)} \cdot \nu \tilde{\varepsilon} \cdot h. \quad (12) $$

From the latter equality, one can state the following proposition, that will be of some (technical) importance in the remainder.

<footnote n="8">⁸High variability of the results with respect to the cost structure is a general feature of matching models. Hence, the assumption of a constant replacement ratio, although not a bad one in the long-run, is not as innocuous as it may appear at first sight. We shall see below some of its implications.</footnote>
**Proposition 2.** For any density $\varphi$ with unbounded support, one has

$$ \frac{\omega\varphi(\omega)}{1 - \Phi(\omega)} \underset{\omega \to +\infty}{\longrightarrow} \eta \in [0, 1). $$

— If $\eta \ge (1-z)/(1-z+z\nu)$, then equation (12) has no solution.

— If $\eta < (1-z)/(1-z+z\nu)$, then that equation defines $w^r/h$ as a derivable function of $p, z, \nu, r$, and $n$. It increases w.r.t. $p, z$ and $\nu$, and it decreases w.r.t. $r$ and $n$. Furthermore, it has strictly positive limits as $p$ tends to 0 and $+\infty$. The former limit is always finite, and the latter is $+\infty$ if and only if $\eta = 0$.

**Proof:** Please consult the appendix. $\square$

We assume throughout the paper that the condition $\eta < (1-z)/(1-z+z\nu)$ is met. It is always true for most ‘usual’ densities (as per instance the exponential and normal densities, for which $\eta = 0$). Moreover, for notational simplicity, we will omit the exogenous arguments of the function defined in the latter proposition, so that we shall write $w^r/h = \omega(p)$.

Interpreting the responses of $\omega$ to shifts in its (implicit or explicit) arguments is straightforward. Anything that makes the unemployed workers better off raises the reservation wage, whereas an increase in $r$ or in $n$ makes the workers more impatient, hence less exacting.

Now, a steady state equilibrium of our model is defined by a pair of equations: the equality of the flows into and out of unemployment, and the condition of free entry on the search market. Since there are $ne^{-nh}$ people surviving school and stepping into unemployment at every date, and $p \cdot \bar{u}$ matches per unit time that correspond to $q \cdot \bar{v}$ hires, steady state on the labor market simply writes down as $p \cdot \bar{u} = q \cdot \bar{v} = ne^{-nh}$ — an equation known as defining a schedule in the $(\bar{u}, \bar{v})$ plane termed ‘Beveridge curve’ in the matching literature. We now focus on the free entry condition, $^{\top}\Pi^v = 0$. From equations (1), (2) and the equivalence of the ($^{\top}$) and ($\bot$) operators, it can be written as:

$$ \tilde{\Pi}^o = \frac{\tilde{\varepsilon} \cdot h - \tilde{w}}{r + n} = \frac{\kappa}{q}, \qquad (13) $$

which in turn can be rearranged, by means of (11), as:

$$ q = \frac{\nu\kappa}{(1-\nu)(1-z)} \cdot \frac{p+z(r+n)}{\omega(p) \cdot h}. \qquad (14) $$

This entails a first relationship between $q$ and $p$, which we shall denote by $q = \mathcal{F}\mathcal{E}_h(p)$. $\mathcal{F}\mathcal{E}$ stands for ‘free entry’, and subscript $h$ reminds that the free entry condition is parameterized
by the level of schooling effort, which we shall make endogenous in the next section. The main properties of the function $\mathcal{FE}_h$ are summarized in the following proposition:

**Proposition 3.** The free entry condition (14) defines $q$ as a derivable function of $p$, $q = \mathcal{FE}_h(p)$. This function has the following properties: $\mathcal{FE}'_h(\cdot) > 0$, and $\mathcal{FE}_h(0) > 0$. Furthermore, a positive shock on $h$ shifts the free entry schedule downwards in the $(p, q)$ plane.

**Proof:** From equation (14), one sees that $\mathcal{FE}'_h(p)$ has the sign of $\omega(p)-[p+z(r+n)]\cdot d\omega/dp$.

Differentiating equation (12) entails:

$$[p + z(r + n)] \cdot \frac{d\omega}{dp} = \frac{\frac{\tilde{\epsilon}}{\omega} - 1}{\frac{\tilde{\epsilon}}{\omega^2} + \left(\frac{\tilde{\epsilon}}{\omega} - 1\right) \cdot \frac{\varphi(\omega)}{1 - \Phi(\omega)}}$$

$\mathcal{FE}'_h(p)$ thus has the sign of

$$\frac{\tilde{\epsilon}}{\omega} + \left(\frac{\tilde{\epsilon}}{\omega} - 1\right) \cdot \frac{\omega\varphi(\omega)}{1 - \Phi(\omega)} - \left(\frac{\tilde{\epsilon}}{\omega} - 1\right) > 0,$$

for $\tilde{\epsilon}/\omega - 1$ is always positive (see the proof of proposition 2 in the appendix). The remainder of the statement is straightforward. □

The downward slope of $\mathcal{FE}_h$ is a feature that was not obvious a priori. Indeed, increasing the workers' matching rate $p$ directly raises their reservation wage, since the function $\omega(p)$ is increasing. This has a twofold effect. On the one hand, it directly makes occupied jobs less profitable on average, by raising the expected wage. This makes the entrepreneurs less willing to open vacant jobs, which makes matches more frequent for those who keep on searching. On the other hand, a higher reservation wage reduces the share of acceptable matches (the size of which is $1 - \Phi \circ \omega(p)$). This tends to diminish $q$. Finally, proposition 3 shows that the former effect always overcomes the latter.

Note that the function $\mathcal{FE}_h$ is also parameterized by $\kappa$, $h$, $z$, $\nu$, $r$, and $n$. A rise in the hiring cost $\kappa$ unambiguously raises $q$ by making vacancies less profitable. A rise in $h$ has the exact opposite effect. A shift on any of the other four parameters affects the reservation wage, thus activating both of the above depicted mechanisms. They work in the same direction in response to a shift on $r$ or $n$. Increasing the discount rate makes all agents more impatient, which reduces the reservation wage and makes the entrepreneurs open less vacant jobs, thus unambiguously raising their hiring rate $q$. A shift in either $z$ or $\nu$, however, puts those mechanisms in conflict, and makes it impossible to determine the sign of the resulting variation in $q$.
All we need by now is a second relationship between $p$ and $q$. Plugging the equilibrium value of the reservation wage (12) into the expression of the individual matching rate (10) yields:

$$p = \mu \left( 1, \frac{\bar{v}}{\bar{u}} \right) \cdot \left[ 1 - \Phi \circ \omega(p) \right]. \qquad (15)$$

The remarkable thing about (15) is that it shows that the equilibrium definition of the individual probability of matching does not depend on the average length of schooling time $h$, although equation (10) showed that individual schooling effort did influence that probability as it was perceived by individuals. This feature is quite satisfactory, since there is no reason to believe that longer studies on average in a given economy should improve the working of its labor market. It nevertheless hinges crucially on the constant replacement ratio assumption. It would have been false if we had kept the unemployment compensation exogenous in equilibrium. Indeed, the reservation wage depends on the unemployed workers' earnings (it is a weighted average of unemployment benefits and the expected wage). Hence, if unemployment income is kept proportional to the average expected wage, then a higher schooling effort, by raising the wage, also makes the unemployed better off and raises their income proportionally. As a result, both 'components' of the reservation wage vary proportionally to the schooling effort, which implies the same proportionality for the equilibrium reservation wage itself.

Steady state on the labor market implies that $p \cdot \bar{u} = q \cdot \bar{v} \Leftrightarrow \bar{v}/\bar{u} = p/q$. This, together with (15), implies:

$$p = \mu \left( 1, \frac{p}{q} \right) \cdot \left[ 1 - \Phi \circ \omega(p) \right]. \qquad (16)$$

This equation yields another relationship between $p$ and $q$, denoted $q = MP(p)$ (where $MP$ stands for 'matching probability'). About this relationship, one has the following:

**Proposition 4.** Equation (16) defines $q$ as a derivable function of $p$, $q = MP(p)$. This function has the following properties: $MP'(\cdot) < 0$, $\lim_{p \to 0^+} MP(p) = +\infty$, and $\lim_{p \to +\infty} MP(p) = 0$.

**Proof:** Equation (16) can be written as:

$$\mu \left( \frac{1}{p}, \frac{1}{q} \right) \cdot \left[ 1 - \Phi \circ \omega(p) \right] = 1.$$

The properties of the contact function $\mu$ imply that the latter equation has one unique solution $q = MP(p)$ for any $p \in \mathbb{R}_+$. The implicit functions theorem thus applies. The sign of $MP'$ is obvious. The limit values come from the properties of function $\mu$ and $\omega$, and the fact that $\Phi \circ \omega(p)$ takes strictly positive and bounded values as $p$ describes $\mathbb{R}_+$. $\square$
Propositions 3 and 4 make the following result obvious:

**Proposition 5.** The model has one unique symmetric steady state equilibrium with exogenous schooling effort, $(p^*, q^*, \bar{u}^*, \bar{v}^*)$, defined by $q^* = \mathcal{F}\mathcal{E}_h(p^*) = \mathcal{M}\mathcal{P}(p^*)$, and $p^* \cdot \bar{u}^* = q^* \cdot \bar{v}^* = n$.

**Proof:** The proposition stems directly from propositions 3 and 4. □

<figure><img src="image_3.png" /></figure>

The unique steady state symmetric equilibrium with exogenous schooling of our model is depicted on figure 1. The labelled arrows represent the shifts of each schedule in response to a positive shock on various parameters (with symbol $\mu$ corresponding to a positive 'neutral' productivity shock on the contact function). All results of comparative statics are quite intuitive. In particular, it is clear that an increase in the average schooling effort will foster employment.

<figure><img src="image_4.png" /><figcaption>Figure 1: symmetric steady state equilibrium with an exogenous schooling effort</figcaption></figure>

Indeed, although it has no impact on the individual probability of matching, it makes the firms more willing to open vacant jobs by making occupied jobs more productive. This tends
to increase the job availability ratio $\bar{v}/\bar{u}$ — that is sometimes referred to as the *labor market tightness* in the matching literature. Hence there will be more matches in equilibrium, with a higher matching rate for the unemployed and a lower rate of returns to vacant jobs for the firms that suffer from the congestion externality inherent to the contact function $\mu$. Unemployment thus drops, which again raises job availability.

We emphasized in the previous section that workers benefited from an investment in schooling through their individual probability of finding a job as well as through their average earnings. Since schooling duration is also a key determinant of equilibrium unemployment, the crux for now is its endogenous determination.

#### 4. STEADY STATE EQUILIBRIUM WITH AN ENDOGENOUS SCHOOLING EFFORT

In this section, we take into account the fact that going to school entails a loss of utility for young agents. Indeed, spending time at school (enjoying zero earnings) implies foregoing wealth. Formally, let $\Omega_{i,t}^s$ be the asset value for agent $i$ of being at school at date $t$. This value behaves according to<fnref n="9" />:

$$ (r+n) \Omega_{i,t}^s = -b_t + \dot{\Omega}_{i,t}^s. \quad (17) $$

Let $h_i$ be the length of agent $i$'s studies. When coming out of school, individual $i$ born at date $t$ enters the labor market as unemployed worker, thus enjoying a wealth $\Omega_{i,t+h_i}^u$. Integrating (17) between $t$ and $t+h_i$ implies that that individual solves the ensuing programme:

$$ \max_{h_i \ge 0} \left\{ \Omega_{i,t}^s = e^{-(r+n)h_i} \cdot \Omega_{i,t+h_i}^u - \int_t^{t+h_i} b_\tau \cdot e^{-(r+n)(\tau-t)} d\tau \right\}. \quad (18) $$

Longer studies imply a higher and more heavily discounted value of unemployment $\Omega_{i,t+h_i}^u$, together with a higher probability of dying at school. Replacing $\Omega_{i,t+h_i}^u$ by its steady state value from (4), one sees that the latter programme boils down to:

$$ \begin{aligned} h_i &= \underset{h_i \ge 0}{\operatorname{argmax}} \left\{ \int_{t+h_i}^{+\infty} (w_i^r - b) \cdot e^{-(r+n)(\tau-t)} d\tau - \int_t^{t+h_i} b \cdot e^{-(r+n)(\tau-t)} d\tau \right\} \\ &= \underset{h_i \ge 0}{\operatorname{argmax}} w_i^r \cdot e^{-(r+n)h_i}, \end{aligned} \quad (19) $$

the first order condition<fnref n="10" /> of which writes down as:

$$ \frac{\partial w_i^r}{\partial h_i} = (r+n) w_i^r. \quad (20) $$

<footnote n="9">⁹We need to assume that “students” have to pay the unemployment tax $b_t$ to ensure the neutrality of that tax.</footnote>

<footnote n="10">¹⁰What we do about the second order condition is cross our fingers.</footnote>
The value of $\partial w_i^r / \partial h_i$ can be taken from the proof of proposition 1. Now, in a symmetric equilibrium where subscripts can be omitted, one can substitute the equilibrium values of $w_i^r$ and $p$ into the expression of $\partial w_i^r / \partial h_i$. This yields the optimal value of the average schooling duration as a function of $p$:

$$h^*(p) = \frac{p[\nu p + (r+n)(1-z+z\nu)]}{(r+n)(r+n+\nu p)[p+z(r+n)]}, \quad (21)$$

which is the value $h$ has to be substituted for in the equilibrium relationships. From (21), one can check that $h^*$ increases (a property that was naturally expected) from $h^*(0) = 0$ to $\lim_{p \to +\infty} h^*(p) = 1/(r+n)$ when $p$ describes $\mathbb{R}_+$. It also clearly appears that $h^*$ decreases w.r.t. $r$ and $n$: more impatient agents are less keen on spending much time at school.

As we already saw, the average duration of schooling is absent from the definition of the equilibrium individual matching rate. Hence, the $MP$ schedule is not affected by the endogeneity of $h$, and it remains the same as in figure 1. The free entry condition (14), however, changes somewhat since parameter $h$ has to be replaced by $h^*(p)$ in its expression. It now defines a function $q = FE(p)$ that can be written as:

$$FE(p) = q = \frac{\nu\kappa}{(1-\nu)(1-z)} \cdot \frac{p+z(r+n)}{\omega(p) \cdot h^*(p)}, \quad (22)$$

and the properties of which are summarized in the following proposition:

**Proposition 6.** The free entry condition (22) defines $q$ as a derivable function of $p$, $q = FE(p)$. Moreover, $\lim_{p \to 0} FE(p) = +\infty$ whenever $z > 0$, and $\lim_{p \to +\infty} FE(p) > 0$.

**Proof:** The first limit value comes straightforwardly from (22), (21) and the fact that $\omega(0) > 0$ (see proposition 2). The second limit value can be obtained by writing the ensuing equivalence:

$$FE(p) \underset{p \to +\infty}{\sim} FE_{\lim_{p \to +\infty} h^*(p)}(p) = FE_{1/(r+n)}(p).$$

The r.h.s. of this relation has a strictly positive limit as $p$ tends to infinity (proposition 3)□

The interesting thing about proposition 6 is that the $FE$ schedule now has decreasing segments. (It can even possibly decrease over the entire real line.) It is henceforth possible that the model exhibits multiple equilibria (see figure 2). The reasons for the existence of downward sloping parts of the $FE$ schedule are easy to find. By raising the average level of human capital, an increase in the individual probability of matching has a twofold effect: on the one hand, it
<figure><img src="image_5.png" /><figcaption>Figure 2: steady state equilibria with an endogenous schooling effort</figcaption></figure>

increases the profitability of occupied jobs, hence encouraging firms to open more vacancies, and on the other hand it raises the bargained wage for obvious reasons, which makes vacant jobs less valuable to entrepreneurs. The former effect dominates for low values of $p$, since $FE$ has to be decreasing over a neighborhood of 0.

Figure 2 exemplifies a situation in which several equilibria coexist. The possibility of such a situation is verified numerically in the appendix. Point A is an equilibrium with high unemployment and a low hiring rate. It also implies a low level of schooling effort. At point C, the contact rate between unemployed workers and vacant job slots is high, which encourages agents to invest in human capital. This in turn fosters the opening of vacancies by firms, which makes the labor market situation even better.

This multiplicity of equilibria can also be interpreted in terms of coordination failures in a statical game (see Cooper and John [1988]). Agents must choose their schooling effort in a non cooperative way. The existence of strategic complementarities in this game — that can be shown to be necessary for the occurrence of multiple equilibria; see Cooper and John [1988] — makes
little doubt. Longer studies on average make the marginal reward of education higher for all individuals, thus creating an incentive for any particular agent *i* to increase her/his own effort $h_i$. That incentive runs through two channels. Firstly, an increase in *h* raises the job availability ratio $\bar{v}/\bar{u}$ (see previous section), which makes all workers meet firms more frequently, thus increasing the frequency of actual matches. Secondly, a better average schooling level means a greater average productivity $\tilde{\varepsilon}$, of which worker *i* captures a share proportional to $h_i$ with an instantaneous probability $p_i$. Analogous arguments show the existence of positive spillovers in this game: a higher average education level raises the reservation wage again through two channels: it raises $p_i$ and the average wage, which makes the unemployment compensation more generous, since we posited a constant replacement ratio<fnref n="11" />. Hence a higher *h* unambiguously improves agent *i*'s welfare. This makes the Pareto ranking of equilibria *A* and *C* obvious: agent unanimously prefer equilibria with a higher level of education.

Now, what figure 2 also shows is that there might be possibilities for a government to keep the economy away from the ‘poor’ equilibrium<fnref n="12" /> *A*, for instance by cutting the hiring costs $\kappa$. Such a cut — note that it needs not be a very big one — could possibly shift the *FE* schedule down to position *FE'*, and force the economy onto a ‘rich’ equilibrium<fnref n="13" /> such as *C'*. Another possibility is to put legal constraints on the level of education. As the equilibrium schooling effort, $h^*(p)$, is lower at point *A* than at point *C*, the government could force the economy to remain at *C* by imposing a minimum age to come out of school, $h_m$, anywhere between the $h^*$ of point *A* and that of point *B*. Again, note that the required increase in the minimum duration of studies needs not be very big, since $h_m$ needs not be very far above the ‘low’ equilibrium level of education to force the economy onto point *C*.

## 5. GROWTH ANALYSIS

Something this paper has ignored so far is the positive external effects associated with education, as emphasized by Uzawa [1965] and Lucas [1988]. A natural extension is to put education-based endogenous growth into the model, a task this section is devoted to.

<footnote n="11">¹¹The latter channel may thus appear less convincing than the former. Both are independant, however, and positive spillovers would be found even under a different assumption about the unemployment benefits.</footnote>

<footnote n="12">¹²Yet Saint-Paul [1996] shows that this is not necessarily the best thing to do when the labor force is heterogeneous with respect to skills, for, as is pointed out by this author, the unskilled unemployment rate increases with the proportion of skilled workers in the economy.</footnote>

<footnote n="13">¹³It is implicitly assumed that our three equilibria have suitable stability properties. More precisely, *A* and *C* need to be unstable, while *B* should be stable. All we can say without a thorough study of the model’s dynamics is that point *B* has counter intuitive comparative static properties, and henceforth cannot be considered a ‘proper’ equilibrium.</footnote>
The per unit time output of any match *ij* is now assumed to be $a_t \cdot \varepsilon_{ij}h_i$, where $a_t$ is a measure of the 'technical state' of the economy, and grows at an endogenous rate, the expression of which will be given later on. To ensure the existence of a balanced growth path, we shall also assume that the cost of hiring is also proportional to the state of the art, and therefore turn $\kappa$ into $\kappa \cdot a_t$. This assumption is somewhat *ad hoc*, albeit quite frequent. (Bean and Pissarides [1993] provide a rationale for it, by saying that the cost $\kappa$ represents the value of leisure foregone by the entrepreneur in job advertising.) Under those assumptions, and restraining ourselves to such a balanced growth path, along which $a_t$ grows at a constant rate $g$, all results of sections 2 and 3 remain valid conditionally on replacing $h_i$ by $a_t \cdot h_i$, $\kappa$ by $\kappa \cdot a_t$, and $r+n$ by $r+n-g$. The latter change is the most important, and is easily interpreted: growth makes agents less impatient, since they now take their income growth rate ($g$) into account when discounting the future. This can be seen when rewriting the first order condition (20) for the education decision of any agent $i$. It turns into

$$ \frac{\partial w_{i,\tau+h_i}^r}{\partial h_i} = (r+n-g) w_{i,\tau+h_i}^r. \qquad (23) $$

Now replacing the (properly amended to make them take account of growth) equilibrium values of the reservation wage and the matching rate $p$ in the latter equation yields the optimal duration of studies in a growth context:

$$ h_g^*(p) = \frac{p[\nu p + (r+n-g)(1-z+z\nu)]}{(r+n-g)(r+n-g+\nu p)[p+z(r+n-g)]} \qquad (24) $$

Going to school implies a 'waste of time', the value of which is measured by the gross discount rate $r+n$. This loss is partially<fnref n="14" /> recovered by the meanwhile increase in the reservation wage, which changes that gross discount rate into a net discount rate, $r+n-g$. More growth thus encourages the agents to undertake longer studies, which in turn reduces unemployment.

Thinking of education as the engine of growth adds a positive feedback between growth and education (and henceforth employment). Following Uzawa [1965] and Lucas [1988], we assume that activity in the educational sector is uniformly diffused over the entire economy, so that the rate of growth of the aggregate stock of 'technical knowledge' is determined by the average schooling effort:

$$ \frac{\dot{a}_t}{a_t} = g \left( \int_{-\infty}^{t} h_{\tau} \cdot ne^{-n(t-\tau)} d\tau \right), \qquad (25) $$

with $g$ some positive and increasing function, and $h_\tau$ the schooling effort of generation $\tau$'s representative agent. On a balanced growth path, the growth rate simply becomes $g(h)$, an

<footnote n="14">¹⁴One must assume $g < r+n$ to ensure the finiteness of the schooling effort.</footnote>
increasing function of the equilibrium schooling effort. Hence, more employment means an
incentive to undertake longer studies, which fosters growth, that in turn encourages education
and employment.

Finally, making the growth rate endogenous adds a channel for strategic complementarities. Indeed, a better average level of education accelerates growth, thereby making individuals stay in school a bit longer. For that reason, it seems fair to bet on the existence of multiple balanced growth paths with an unambiguous Pareto ranking.

6. CONCLUSION

This paper examines the nexus between the educational system, the labor market, and
economic growth. It shows that, although influencing individual job opportunities, the level of
education undertaken by workers does not affect the equilibrium matching rate. It nevertheless
influences labor market equilibrium by making labor more productive and at the same time more
expensive. This ambiguous influence may give birth to multiple steady state equilibria of the
model, on the ‘best’ of which we show that it is possible for a government to keep the economy.
We finally make the growth rate endogenous by introducing human capital externalities into
the model. We show the existence of a positive feedback between employment, education and
growth.

The model leaves a number of questions open, some of which might be of interest for further
research. In particular, education in this model is only associated with positive externalities, a
feature that rules out inefficient *over*-education. Such a possibility would arise in a world where
workers would be in competition with one another when applying for jobs. Assume for instance,
as in the original model of Diamond [1982], that the number of jobs available is held constant
instead of being allowed to vary optimally. In such a context, a firm could find it worthwhile,
when meeting a worker, to refuse the match and wait for a more productive partner instead of
wasting a valuable vacant job (which has become a scarce resource) with a low-skilled worker.
Longer studies on average would then act as a negative externality for individuals, by diminishing
their chances to be accepted by firms.

REFERENCES

Bean, C. and C. A. Pissarides, *Unemployment, Consumption and Growth*, *The European Economic Review*, 1993, *37*, 837–59.
Blanchard, O. J. and S. Fischer, *Lectures on Macroeconomics*, Cambridge, MA.: The MIT Press, 1989.

Cahuc, P. and P. Michel, Minimum Wage, Unemployment and Growth, *The European Economic Review*, 1996, **40**, 1463–82.

Cooper, R. and A. John, Coordinating Coordination Failures in Keynesian Models, *Quarterly Journal of Economics*, 1988, **103** (3), 441–63.

Diamond, P. A., Wage Determination and Efficiency in Search Equilibrium, *Review of Economic Studies*, 1982, **44**, 217–27.

Fershtman, C., K. M. Murphy, and Y. Weiss, Social Status, Culture and Economic Performance, *Journal of Political Economy*, 1996, **104** (1), 108–32.

Laing, P., T. Palivos, and P. Wang, Learning, Matching and Growth, *Review of Economic Studies*, 1995, **62**, 115–29.

Lucas, R. E., On the Mechanics of Economic Development, *Journal of Monetary Economics*, 1988, **22**, 3–42.

Pissarides, C. A., *Equilibrium Unemployment Theory*, Oxford: Basil Blackwell, 1990.

Saint-Paul, G., Unemployment, Wage Rigidity, and the Returns to Education, *The European Economic Review*, 1994, **38**, 535–43.

—, Are the Unemployed Unemployable?, *The European Economic Review*, 1996, **40**, 1501–19.

Uzawa, H., Optimum Technical Change in an Aggregative Model of Economic Growth, *International Economic Review*, 1965, **6** (1), 18–31.
# APPENDIX

## A. PROOF OF PROPOSITION 1

First note that $\perp\varepsilon_i$ is defined by:

$$ \perp\varepsilon_i = E_{\varepsilon}(\varepsilon_{ij} | \varepsilon_{ij} \ge w_i^r / h_i) = \frac{\int_{w_i^r / h_i}^{+\infty} \varepsilon \, d\Phi(\varepsilon)}{1 - \Phi(w_i^r / h_i)}. $$

Replacing that definition and equation (10) into equation (9) yields an equation in the sole variable $w_{i,t}^r$, that can be written as:

$$ \left\{ r + n + \nu\mu \left( 1, \frac{\bar{v}}{\bar{u}} \right) \cdot \left[ 1 - \Phi \left( \frac{w_i^r}{h_i} \right) \right] \right\} \cdot w_i^r - \nu\mu \left( 1, \frac{\bar{v}}{\bar{u}} \right) h_i \cdot \int_{w_i^r / h_i}^{+\infty} \varepsilon \, d\Phi(\varepsilon) = (r+n)\bar{z}. $$

The derivative of the latter equation's l.h.s. with respect to $w_i^r$ is worth $r+n+\nu p_i > 0$. Hence, that l.h.s. increases from some finite negative value to $+\infty$ as $w_i^r$ describes $\mathbb{R}_+$. The above equation thus has one unique solution, to which the implicit functions theorem applies. Simple algebra shows that:

$$ \frac{\partial w_i^r}{\partial h_i} = \frac{\nu p_i \cdot \perp\varepsilon_i}{r + n + \nu p_i} > 0. $$

Furthermore, one sees from (10) that $\partial p_i / \partial h_i$ has the sign of $-\partial(w_i^r / h_i) / \partial h_i$, which can easily be derived:

$$ \frac{\partial}{\partial h_i} \left( \frac{w_i^r}{h_i} \right) = \frac{\nu p_i \cdot \perp\varepsilon_i h_i - w_i^r (r + n + \nu p_i)}{h_i^2} = - \frac{(r + n) \bar{z}}{h_i^2} < 0. $$

(The last equality holds by virtue of (10).) Thus, $\partial p_i / \partial h_i > 0$. All the remaining signs are straightforward.

## B. PROOF OF PROPOSITION 2

Let $\omega = w^r/h$. $\tilde{\varepsilon}$ is defined by

$$ \tilde{\varepsilon} = \frac{\int_{\omega}^{+\infty} \varepsilon \, d\Phi(\varepsilon)}{1 - \Phi(\omega)}. $$

It is obvious that

$$ \forall \omega \in \mathbb{R}_+^*, \quad \frac{\tilde{\varepsilon}}{\omega} = \frac{\int_{\omega}^{+\infty} \frac{\varepsilon}{\omega} d\Phi(\varepsilon)}{\int_{\omega}^{+\infty} d\Phi(\varepsilon)} \ge 1. $$

One can check that

$$ \frac{d}{d\omega} \left( \frac{\tilde{\varepsilon}}{\omega} \right) = \frac{\varphi(\omega)}{1 - \Phi(\omega)} \cdot \left( 1 - \frac{\tilde{\varepsilon}}{\omega} \right) - \frac{\tilde{\varepsilon}}{\omega^2} < 0. $$
Since the function $\tilde{\epsilon}/\omega$ is strictly decreasing and bounded from below by 1, it has some limit $\lambda \ge 1$ when $\omega$ tends to infinity. Now applying l'Hôpital's rule to the fraction defining $\tilde{\epsilon}/\omega$ yields:

$$ \lambda = \lim_{\omega \to +\infty} \tilde{\epsilon}/\omega = \lim_{\omega \to +\infty} \left( 1 - \frac{1 - \Phi(\omega)}{\omega\varphi(\omega)} \right)^{-1}. $$

Hence we can define

$$ \eta = \lim_{\omega \to +\infty} \frac{1 - \Phi(\omega)}{\omega\varphi(\omega)} = 1 - \frac{1}{\lambda} \in [0, 1). $$

Hence the first part of the proposition. Furthermore, one easily sees that:

$$ \lim_{\omega \to 0^+} \tilde{\epsilon}/\omega = +\infty. $$

Equation (12) may be rearranged as:

$$ \frac{\tilde{\epsilon}}{\omega} = 1 + \frac{(1-z)(r+n)}{\nu[p+z(r+n)]}. $$

The r.h.s. of the latter relation clearly decreases w.r.t. $p$ and $z$, and increases w.r.t. $\nu$, $r$, and $n$. It describes $(1, 1 + (1-z)/(\nu z)]$ as $p$ describes $\mathbb{R}_+$. Now, if $\lambda = 1 - 1/\eta < 1 + (1-z)/(\nu z)$, then the implicit functions theorem will apply and complete the proof of the proposition straightforwardly. In the opposite case, equation (12) has no solution. $\square$

### C. MULTIPLE EQUILIBRIA

The situation depicted on figure 3 was obtained with the ensuing specifications: $r = 0.1$, $n = 0.05$, $\nu = 0.5$, $z = 0.6$, $\kappa = 0.0012$, $\mu(\bar{u}, \bar{v}) = 0.0075 \cdot (\bar{u}^{1-1/5} + \bar{v}^{1-1/5})^{5/(5-1)}$, and $\varphi(\epsilon) = 10 \cdot e^{-10 \cdot \epsilon}$. The solid line is the *FE* schedule, and the dashed line is the *MP* schedule.
<figure><img src="image_6.png" /><figcaption>Figure 3: multiple equilibria</figcaption></figure>