DP2023-13

Impacts and Distribution of Premiums
from Temporal Social Networks
across Generations*

RIEB Junior Research Fellow

Yoshitaka OGISU

June 21, 2023
* This Discussion Paper won the Kanematsu Prize (FY 2022).

Impacts and distribution of premiums from temporal social
networks across generations∗
Yoshitaka Ogisu†
Graduate School of Economics, Kobe University
Kobe, Japan
June 21, 2023

Abstract
Social networks certainly play an important role in labor market outcomes. In particular, the structures aﬀect inter-group inequality via referral hiring. Through the
network eﬀects, while workers surely get premiums from the group to which they belong, they may get premiums or penalties from other groups than their own. Youngworkers do not obtain suﬃcient network premiums since referrals cannot be used well
due to the higher unemployment rates of their friends. As time goes by, the network
structure of each generation of course changes. In other words, not only premiums
from their own network group but also those from the other network groups, or the
spillovers from other generations, change over time. However, these changes in intraand inter-group network eﬀects have been rather overlooked so far. In this paper, we
compute the network premiums for each generation in a search and matching model,
and clarify which generation beneﬁts the most from time-varying networks called temporal networks. New connections are generated proportional to the number of friendsof each worker over time, while the existing connections are broken at a constant rate.
Under this setting, workers get premiums or penalties depending on their network
structures. On average, workers receive premiums from the overall network eﬀects although they incur penalties from their network structures in wage and unemployment
rates.
JEL Classiﬁcation: E24, J31, J64;
Keywords: Referral hiring; temporal network; network structure; intergenerational
inequality.

∗

This work won the Kanematsu Prize from Research Institute for Economics and Business Administration (RIEB), Kobe University in 2022. I acknowledge ﬁnancial support from the Kanematsu Prizeand the Doctoral Student Fellowship at Kobe University. I am grateful to Makoto Masui, Teruyoshi
Kobayashi, Tamotsu Nakamura, and reviewers of the Kanematsu Prize for their constructive comments
and discussions.
†E–mail: yoshitaka.ogisu@gmail.com

1

Introduction

Referral hiring plays a non-negligible role in labor market outcomes, especially in determining inequality among workers. Many workers ask their friends for vacant positions.
Thus, those who have many friends have high job-ﬁnding probabilities and accordingly
strong outside options in wage bargaining leading to high wages, whereas those who havea few friends do not (Fontaine, 2008; Ioannides and Soetevent, 2006). Workers beneﬁt
from referral hiring through their social networks in which workers and friendships arenodes and edges, respectively.
Social networks surely change over time, and those time-varying networks are called
temporal networks (Holme and Saramäki, 2019). This idea applies to a worker network
since new friendships are formed and existing ones are broken up. If a worker is friendly,
and has many friends, he is easy to make new friends, otherwise, he is diﬃcult to makenew ones, which is measured by degree correlations of assortativity (Jackson and Rogers,
2007; Newman, 2002, 2003). If this is the fact for all workers, then social networks are
structured in such a way that a small number of people become to have a large numberof friends as time goes by. This leads to structural diﬀerences in the social networks
across generations, say, the young and old generations, meaning that workers of diﬀerent
generations (or ages) can have diﬀerent premiums from their social networks.
The labor market with referral hiring using social networks has been analyzed sinceseminal works by Rees (1966), Granovetter (1973), and Boorman (1975) because referral
hiring is one of the typical methods used by workers (Holzer, 1988; Pellizzari, 2010).1 An
analytical framework is established by Calvó-Armengol and Zenou (2005) and Fontaine(2007) in which referral hiring is incorporated as social networks into a search and matchingmodel proposed by Diamond (1981), Mortensen and Pissarides (1994), and Pissarides(2000).
In search and matching models, social networks are often assumed to have a simple
structure (e.g. a regular network in which all nodes have the identical number of edges) foranalytical tractability (e.g. Calvó-Armengol and Zenou, 2005; Fontaine, 2007; Galenianos,2014). However, it is well known that the network structure itself, e.g. how edges connect
nodes in a network, and how workers diﬀer in the number of friends, has a signiﬁcant
impact on the unemployment probability of a worker, as Calvó-Armengol and Jackson(2004) and Tassier and Menczer (2008) have shown using small networks.
To clarify the characteristic of a network structure in search and matching models,Ioannides and Soetevent (2006) uses the idea of degree distribution. The degree of anode in a network is the number of connections each node has to other nodes, and hence
the degree distribution is deﬁned as the probability distribution of the degrees over the
network.2 The degree distribution in a social network corresponds to the distribution of
the number of friends that each worker has. With degree distributions, diﬀerences in social
network structures lead to diﬀerences in congestion for sharing information about vacant
job positions (Ogisu, 2022). Since social networks in the previous models are static, wedevelop a theoretical model with temporal networks in which network structures can be
diﬀerent among generations.
In this paper, we attempt to capture generation-dependent premiums (or penalties)caused by temporal social networks. We employ a search and matching model with overlapping generations, called life-cycle search models (e.g. Chéron et al., 2013; Fujimoto,2013), and incorporate temporal networks into it. First, we construct the model with sim1
2Comprehensive surveys are provided by Ioannides and Loury (2004); Topa (2011).
See Barabási (2013) and Newman (2018) for basics about degrees and degree distributions.

1ple social networks, random regular networks, and analytically demonstrate the existenceof equilibrium. Second, we extend the model to include temporal networks generated bya ﬁtness model (e.g. De Masi et al., 2006; Kobayashi et al., 2019). In a ﬁtness model,
each node has an activity parameter, and the probability of creating an edge between two
nodes depends on the parameters. To describe degree correlations in a social network, we
set the activity parameters proportional to the number of friends the worker has.
The main results in the paper are summarized as follows. First, workers can receive
positive premiums from their social networks. Judging from the Japanese data, the premiums in wage and unemployment rates are up to about 1.6%pt (measured by the averagewage) and 0.2%pt, respectively, although the premiums are partly oﬀset by structures ofsocial networks up to 0.3%pt and 0.03%pt, respectively.
Second, young workers cannot beneﬁt from their own social networks well even if
they have relatively good social network structures. This is because the probability that
their friends are also unemployed is high, and hence young workers are diﬃcult to obtain
valuable information from their friends.
Third, an increase in the network connectivity of some generations (e.g. average number of friends) can lead to a decline in the unemployment rates not only in that generation
but also in the other generations. When the network connectivity in some generation
increases, it improves the matching probability via referrals in that generation, which inturn brings about an increase in the number of new entry ﬁrms. If the number exceeds
additional matches made by workers in the generation in which the change takes place atﬁrst, the vacancy rate increases in the network as a whole, leading to improved matching in
the labor market. As a result, the other generations can also obtain additional matches. If,
however, the number of entry ﬁrms is not suﬃcient, then the opposite case is also possiblethat the vacancy rate decreases and the unemployment rates of the other generations rise.
The results of our numerical calculations conﬁrm that an increase in network connectivityof the middle aged group (40 y.o.) raises the unemployment rate in the other generations.
This paper contributes to the existing literature threefold. The ﬁrst is to presenta theoretical analysis to explore the eﬀects of referral hiring using social networks (e.g.
Alaverdyan and Zaharieva, 2022; Cahuc and Fontaine, 2009; Galenianos, 2013; Galeotti
and Merlino, 2014; Horvath and Zhang, 2018; Zaharieva, 2018). In the literature, someworks have investigated eﬃciency losses (e.g. Cahuc and Fontaine, 2009; Galenianos, 2014;
Horváth, 2014), while some examine the inequality caused by referral hiring (e.g. CalvóArmengol and Jackson, 2004, 2007; Galenianos, 2021; Horváth, 2014; Igarashi, 2016; Stupnytska and Zaharieva, 2015; Zaharieva, 2013). Our analysis is in the line with the latter.
It is particularly diﬀerent from the former in using a life-cycle model to introduce generational diﬀerences among worker networks to capture generation-dependent premiums.
The second is examining inequality across generations in a life-cycle search model(e.g. Chéron et al., 2011, 2013; Esteban-Pretel and Fujimoto, 2014; Fujimoto, 2013; Hahn,
2009). We explicitly introduce referrals with social networks into the model in a tractablemanner.
The third is to focus on the role of temporal networks (e.g. Holme and Saramäki, 2019;
Kobayashi and Takaguchi, 2018; Kobayashi et al., 2019). In our model, social networks
are exogenously given based on a ﬁtness model that is frequently used in recent studies
in network science. Since little is known about the impact of temporal networks on thelabor market, this paper is one of the important applications of social networks using asearch and matching model.
The rest of the paper is constructed as follows. Sec. 2 describes the basic model with alife-cycle search and matching model. With assumptions for analytical tractability, Sec. 3

2

shows the existence of equilibrium and deals with comparative analysis showing that an
expansion in the eﬀect of network connectivity has spillover eﬀects. With parameters
calibrated to ﬁt the Japanese data, Sec. 4 presents the algorithm generating temporalsocial networks and identiﬁes premiums and/or penalties from social networks numerically.
Sec. 5 summarizes the results and describes research avenues.

2

Model

Time is a discrete and inﬁnite horizon, and the economy is in a steady state. Workers
who are risk neutral enter the market for generation 0 at ﬁrst, and leave the market when
they reach the end of the last generation, ḡ. Thus, there are generations of workers granging from 0 to ḡ in the economy. Assume that workers in each generation make up asocial network with a suﬃciently large number of workers, N , which is constant and thesame for all generations. For simplicity, there is no friendship between social networks in
diﬀerent generations. A worker who is unemployed searches for a job and produces home
production b > 0, or one who is matched with a job supplies a unit of labor and receives
wage wg .
There is an inﬁnite number of jobs which is ready to enter the market. If a job isvacant, the job is posted in the labor market with cost c > 0. If a worker arrives at
the job, the job becomes a matched job and produces yg unit of general good and pays
wage wg to the matched worker of generation g. Firms are destructed with a constantprobability of δ ∈ (0, 1) for each end of the time period. Variables c, yg , δ are exogenous
parameters, although wg is determined endogenously at the equilibrium.
Let pg be the (aggregate) job arrival rate for a worker in generation g. Following alife-cycle search model (e.g. Esteban-Pretel and Fujimoto, 2014; Fujimoto, 2013), valuefunctions are given by, for generation g ∈ {0, 1, · · · , ḡ − 1},
Wg = wg + β[(1 − δ)Wg+1 + δUg+1 ],(1)

Ug = b + β[pg Wg+1 + (1 − pg )Ug+1 ],(2)

Jg = yg − wg + β[(1 − δ)Jg+1 + δV ],(3)

for the oldest generation ḡ,Wḡ = wḡ ,(4)

Uḡ = b,(5)Jḡ = yḡ − wḡ + βV,(6)where β ∈ (0, 1) is the discount factor, and Wg , Ug , Jg represent the value functions of aworker of generation g who has the matched job, a worker of generation g who does nothave a matched job, and a job which is matched to a worker of generation g, respectively.
V denotes the value function of a vacancy job, not matched with any workers, and it
is given byḡ
XJg λg + (1 − q)V  ,(7)
V = −c + β qP

g=0

where q ≡ g pg ug /v(ḡ
P + 1) is the average arrival rate of a worker in generation g to thejob, and λg ≡ pg ug / g′ pg′ ug′ is the probability that the matched worker is in generation
g.

3

(b)

(a)
Match with job 𝛼𝛼

A

𝛾𝛾𝑣𝑣

Hold job 𝛽𝛽

B

𝛾𝛾𝑣𝑣

Do not match

C

Labor market

D

𝛼𝛼

𝑃𝑃𝑔𝑔

B
𝛽𝛽

Pass job 𝛽𝛽 to arandomly select friend

𝑃𝑃𝑔𝑔

A job comes with prob. 𝑃𝑃𝑔𝑔
along with each edge.

Take expectation over 𝑙𝑙, 𝑘𝑘

𝑃𝑃𝑔𝑔

E Employed with prob 1 − 𝑢𝑢𝑡𝑡

Hold job info. with prob. 𝛾𝛾𝛾𝛾

𝑙𝑙 unemployed (exc. D) of 𝑘𝑘
friends with prob. 𝑓𝑓𝑔𝑔 (𝑘𝑘)C

C chosen with prob.

Wait referrals

11+𝑙𝑙

Employed
Unemployed

Fig. 1. Schematic of matching procedures. (a) A job arrives at a worker with probability γv. When
Worker B is employed and gets a vacancy, Worker B passes it to one of his friends who is
unemployed. If Worker B has two friends, a vacancy is passed to Worker C with probability
1/2. (b) Referral procedure. Worker D has 3 edges in this case and receives a vacancy fromeach friend with probability Pg . If Worker E has k friends and l of k friends are unemployedexcluding worker D, E chooses D with probability 1/(l + 1). We sum up all possible caseswith the probability taking k, l to get the expected probability that a vacancy goes from E
to D, which is Pg .

The matching procedure primarily follows the setting in Calvó-Armengol and Zenou(2005). Vacancies are made available to workers through the market or their social networks. pM and pRg denote the arrival rate via the market and the conditional probabilityof the job arrival rate by referral, respectively. The aggregate job arrival rate, pg , is givenby
pg = pM + (1 − pM )pR(8)g.
We note that the matching function induced by pg is not a constant return scale in ug andv since we employ the same matching procedure in Calvó-Armengol and Zenou (2005).
We assume that the matching probability in the market is such that the linear functionof vacancy rate v, meaning thatpM = γv,(9)where γ > 0 is a parameter.
In the search and matching model with referral hiring, it is assumed that vacancy
information goes not only to unemployed workers, but also to employed workers (Fig. 1a).
Every worker can hold only one vacancy information and if multiple vacancies are madeavailable to a worker, one is chosen by the worker at random and the others vanish. If
an employed worker receives a vacancy, he passes it to one of his unemployed friends
chosen at random. When unemployed workers cannot get job information in the labormarket, they wait for their friends to pass on information regarding a job position. The
referral procedure is described in Fig. 1b. We ﬁrst deﬁne Pg as the probability that an
unemployed worker receives a vacancy from an arbitrary one of his friends. Starting froman unemployed worker (Worker D in Fig. 1b), one of his friends (Worker E) is employed4with probability 1 − ug .3 Worker E receives information about a vacancy in the labormarket with probability γv. If Worker E has a vacancy, she sends it to one of her friends
with probability ψ ∈ [0, 1]. When Worker E has l unemployed of k friends except forWorker D, Worker E passes Worker D a vacancy with probability 1/(l + 1). Furthermore,combinations of selecting l unemployed workers from k friends are given by a binomial
distribution with probability ug . Lastly, we get

k−1 
XX
k−1
1 l
Pg = ψ(1 − ug )γv
u (1 − ug )k−l−1 fg (k)
l
l+1 g
k

= ψ(1 − ug )γv

l=0

X 1 − (1 − ug )kkugkfg (k),

where fg (k) is the probability that a worker of generation g has k friends.4 Thus, at least
one vacancy information is made available to a worker having k friends with probability
1 − (1 − Pg )k .
The average arrival rate via referral for generation g is given byiXhpR
1 − (1 − Pg )k fg (k), g ∈ {0, 1 · · · , ḡ − 1},
(10)
g =
k

and pRḡ = 0. This is basically the same formulation as in Calvó-Armengol and Zenou(2005). Finally, if we know the degree distribution fg (k) for all g and assume u0 = 1, theequilibrium can be deﬁned.
Deﬁnition 1. An equilibrium is deﬁned such that:
1. The unemployment rate for each generation is determined by the following rules:
fg (k) ∀g : given,u0 = 1,(11)

ug+1 = (1 − pg )ug + δ(1 − ug )∀g ∈ {0, · · · , ḡ − 1},

where pḡ = 0.
2. Wage is determined by Nash bargaining with worker’s bargaining power η ∈ (0, 1),
which implies
wg = arg max (Wg − Ug )η (Jg − V )1−η .
(12)w

3. Free entry condition is satisﬁed:
V = 0.

(13)

Let the match surplus be Sg ≡ Wg − Ug + Jg − V . From the Nash bargaining problemof (12), we obtain,
Wg − Ug = ηSg ,Jg − V = (1 − η)Sg .
3
We consider ug as the probability that a worker chosen from the worker pool of generation g isunemployed. That is one of the approximations by mean-ﬁeld.
4
In this formulation, we implicitly assume that social networks have a locally tree-like structure inwhich the local cycle by three nodes can be ignored(e.g. Molloy and Reed, 1995; Newman, 2018).

5

Combining them with Eq.(4)-(6) and free entry condition (13), we obtain,
Wḡ = wḡ = ηyḡ + (1 − η)b,

(14)

Jḡ = (1 − η)(yḡ − b).

(15)

The match surplus is determined such that
Sg = yg − b + β[1 − δ − pg η]Sg+1 ,∀g ∈ {0, · · · , T − 1},

Sḡ = yḡ − b.

(16)
(17)

At the equilibrium, by Eq.(7) and free entry condition (13),c(ḡ + 1)= G(v),β(1 − η)where G(v) ≡

3

(18)

Pḡ

g=0 ug pg Sg /v.

Analytical framework in regular networks

For the ﬁrst step, we solve the model analytically with some assumptions that simplify
the analysis. In this section, we suppose that yg = y > b ∀g, and each network structure
is a random regular network with degree k̄g . In that case, the degree distribution forgeneration g is given by(
1 if k = k̄g ,fg (k) =(19)
0 otherwise.

3.1

Equilibrium

With the assumption of random regular networks, Pg and pR
g can be rewritten as
Pg = ψ(1 − ug )γv

1 − (1 − ug )k̄g,
k̄g ugkpR
g = 1 − (1 − Pg ) ,for g ∈ {0, · · · , ḡ − 1}.

In this simple case, we derive the next proposition.
Proposition 1. Assume that social networks are regular networks for every generation,i.e., degree distributions are given by (19). Then an equilibrium exists if parameters satisfy
the conditionḡ γβ(1 − η)(y − b) 1 − [β(1 − δ − η)]ḡ−g+1
γβ(1 − η)(y − b) X 1 − [β(1 − δ)]ḡ−g+1.
<1<c(ḡ + 1)
1 − β(1 − δ − η)c(ḡ + 1)
1 − β(1 − δ)
g=0

(20)
Proof. See Appendix A.
There are two remarks for condition (20). First, the existence of the equilibrium isirrelevant to the network parameter k̄g , which is the same as in Calvó-Armengol and
Zenou (2005). The condition only needs information at the end points of v, meaning thatv = 0, 1/γ. For the existence of the equilibrium, the key is not social networks, and thisis held in the model including the generational structure. At these two end points, the
6

(a)

Pḡ

(b)

g=0 S gG(v)γc(ḡ+1)β(1−η)

γS 0
v=0

v∗

1
γ

0

v

v

1
γ

Fig. 2. Graphical image of the uniqueness of the equilibrium. (a) When v = 0, the slope of G(v)is negative for v ∈ (0, 1/γ), and the unique equilibrium exists if Eq. (20). (b) If v > 0, theslope of G(v) in v ∈ (v, 1/γ) is negative but ambiguous in v ∈ (0, v]. When v < v, theuniqueness of the equilibrium is not assured.

job arrival rate in the labor market to 0 and to 1, respectively, meaning that no one canﬁnd jobs and anybody can ﬁnd jobs in the labor market. Therefore, nobody can rely onreferrals in these polarized cases.
Second, the job opening cost must be in the moderate value range. If job opening
cost c is relatively high, the ﬁrst inequality is held. However, if c is too high, the system
violates the second inequality. As a result, the job opening cost needs to remain at amoderate level for the equilibrium to exist. If the job opening cost is too low, inﬁnite jobs
keep entering the market since jobs expect to make proﬁts, while if it is too high, jobs
leave the market which is no longer attractive.
P
At extreme points of v, G(v) approaches two real values, denoting γS 0 and γ ḡg=0 S g ,although its slope between these points is ambiguous. However, we can prove that theslope is negative above a certain level of v, which is summarized in the next lemma.
Lemma 1. For any parameter sets, there exists v < 1/γ such that for v ∈ (v, 1/γ),
∂G(v)
< 0.
∂v

(21)

Proof. See Appendix B.
Moreover, following discussions in this section, we make the next assumption.
Assumption 1. The parameter sets are chosen such that v = 0.
We note here that Assumption 1 is held under the parameters calibrated in Sec. 4.
Combining Lemma 1 and Assumption 1 leads to the next proposition.
Proposition 2. Assume v = 0. Then Eq. (20) is the necessary and suﬃcient condition
for the existence of the unique equilibrium.

7

Proof. By Assumption 1 and Lemma 1, G(v) is monotonically decreased in v andlim G(v) = γ(y − b)

v→0

ḡ
X
1 − [β(1 − δ)]ḡ+1

,

(22)

1 − [β(1 − δ − η)]ḡ+1.
1 − β(1 − δ − η)

(23)

g=0

lim G(v) = γ(y − b)v→ γ1

1 − β(1 − δ)Therefore, whenlim G(v) <v→ γ1c(ḡ + 1)< lim G(v),β(1 − η) v→0

the unique equilibrium exists and this condition is identical to that in Eq. (20) (also see
Fig. 2).

3.2

Spillover eﬀects from network connectivity

In this model, the relationship between the number of friends (k̄g ) and matching probability via referral (pRg ) draws an inverse-U shape (see Proposition 1 (i) in Calvó-Armengol
and Zenou, 2005). The crucial diﬀerence from Calvó-Armengol and Zenou (2005) is that
network expansion of a generation aﬀects the job arrival rate via referral not only of thegeneration, but also of the other generations. We deﬁne k̂g as the k̄g at which pR
g is maximized. Then, at k̄g = k̂g , pg is also maximized. For simplicity, we tentatively treat k̄g as
a real number, as Calvó-Armengol and Zenou (2005) do. We obtain the following result.
Proposition 3. Given v;(i) for τ ∈ {g + 1, g + 2 · · · , ḡ},k̄g ⋛ k̂g =⇒

∂uτ
⋛ 0,
∂ k̄g

(24)

k̄g ⋛ k̂g

=⇒

∂pτ
⋚ 0,
∂ k̄g

(25)

k̄g ⋛ k̂g =⇒

∂Sτ
⋛ 0.
∂ k̄g

(26)

(ii) for τ ∈ {g, g + 1, · · · , ḡ},(iii) for τ ∈ {0, 1, · · · , ḡ},

Proof. See Appendix C.
Regarding Proposition 3, an increment in k̄g means that all workers gain additional
friendships when they become generation (or age) g. Therefore, if the increment in k̄g < k̂g
improves the job arrival rate for workers in generation g, it improves the unemployment
rate and the job arrival rate of the generation in the following periods. For k̄g > k̂g ,however, pg decreases as k̄g increases, resulting in congestion eﬀects stated in CalvóArmengol and Zenou (2005). When the network connectivity is too high, the probabilityof an unemployed worker getting multiple vacancies is also high, which is congestion in
sharing information.

8

With Proposition 3, the equilibrium eﬀects can be obtained. The keys are shifts in
G(v) and Beveridge curves. First, based on Eq. (24)-Eq. (26), an increment (or decline)in k̄g causes a shift in G(v) and moves the equilibrium vacancy rate also since
∂G(v)
∂ k̄g
=

g−1
X
uτ pτ ∂Sτ
τ =0

v

ug
+
v
∂ k̄g



∂Sg
∂pg
Sg + pg
∂ k̄g
∂ k̄g





ḡ
X
1 ∂uτ
∂pτ
∂Sτ
+
pτ Sτ + uτSτ + uτ pτ
.
v ∂ k̄g∂ k̄g∂ k̄gτ =g+1(27)
The sign of Eq. (27) is ambiguous, and consequently, it is impossible to state the direction of the change in the new equilibrium vacancy rate. If the equilibrium vacancy rateincreases, which means that additional jobs enter the market because of an improvement
in worker arrival rate, the labor market becomes loose (and vice versa).
Second, for generation τ > g, the Beveridge curves shift downward due to an increasein k̄g (Eq. 24) while for other generations, they remain constant. Therefore, if the vacancy
rate increases by a change in k̂g , unemployment rates for all generations decrease. By
contrast, if the vacancy rate decreases due to the change, many generations incur negative
impacts on unemployment rates except for the generations for which the Beveridge curves
shift enough to oﬀset the decrease in unemployment rates.
To summarize, we see lead to three possible eﬀects on uτ for each τ ∈ {0, · · · , ḡ} atthe equilibrium via the change in the increment in k̄g which we term spillover eﬀects.
Spillover eﬀects can be decomposed into direct and indirect eﬀects. An increment ink̄g directly decreases in unemployment rates through shifts in Beveridge curves for some
generations (direct eﬀects) while it indirectly changes in unemployment rates for all generations through a change in vacancy rate (indirect eﬀects). Direct eﬀects surely decrease
unemployment rates although indirect eﬀects can increase those. As a result, the overall
spillover eﬀect for each generation is determined by the relative size of the two eﬀects.
Fig. 3 illustrates the eﬀects from the increment in friendships in generation g fromk̄g = κ to k̄g = κ′ . Suppose that the equilibrium vacancy rate v ∗ is achieved with k̄τ = κ ∀τat ﬁrst and κ < κ′ < k̂τ . If the increment in k̄g leads to a shift of G(v) to the right (Fig. 3a),the vacancy rate moves from v ∗ to v ′ , which is indirect eﬀects. Due to the increase in thevacancy rate, the unemployment rate for every generation decreases. In this case, theintuition of the spillover eﬀect is very simple. When workers in a generation gain new
friendships, the referral probability in the generation increases. This improves the jobworker matching probability and encourages jobs to enter the market. As a result, theaggregate matching probability is improved and the other generations also ﬁnd it easy toﬁnd jobs. The beneﬁts from the increase in k̄g are larger in older generations since theBeveridge curves of generations τ > g shift down (direct spillover eﬀects) although thoseof the other generations do not (remember Proposition 3 (i)).5
Figs. 3b and 3c reveal that G(v) shifts to the left. In this case, the increment in k̄g rises
unemployment rates for all generations along with the prior Beveridge curves. However,this change shifts the Beveridge curves downward for generations above g, leading to adecrease in unemployment rates for these generations. For a generation older than g, if
the latter eﬀect exceeds the former, in other words, the direct spillover eﬀect is larger
than the indirect spillover eﬀect, the generation can achieve a lower unemployment rate
than before (Fig. 3b), otherwise, the unemployment rate rises (Fig. 3c). For both cases of
5

The slopes of the Beveridge curves are negative except for generation 0. See discussions in Appendix

B.

9

(a)

(b)

(c)

G(v)

←
c(ḡ+1)β(1−η)→←uτ (v)

gen.g

u∗g+m

gen.g

←

←

gen.g + m

gen.g + m

gen.g

←
u000
g+m

gen.g + m

u00g+m

u0g+m
0

v∗ v0

1
γ

v 00 v ∗

0

1
γ

0

v 000

v∗

v

Fig. 3. Three possible spillover eﬀects from increment k̄g . m > 0 is assumed and Beveridge curves
only for generation g and g + m are shown and the others are omitted. (a) G(v) shift tothe right, leading to lower unemployment rates for all generations. (b) G(v) shift to theright, leading to lower and higher unemployment rates for generation τ > g and g ≤ τ ,respectively. (c) G(v) shift to the right, leading to higher unemployment rates for all
generations.

Fig. 3b and 3c, the young generations incur negative spillover eﬀects in total, since they
do not beneﬁt from direct spillover eﬀects.
In this section, we ﬁnd that the increase in the number of friends in a social network
has spillover eﬀects with a simple network structure, a random regular network, for social networks. Although the simple network structure simpliﬁes the analysis, it cannot
capture the eﬀects of a network structure. It is known that the network structure canaﬀect the unemployment rates and wages for social groups (e.g. Ogisu, 2022; Tassier andMenczer, 2008). In the next section, we incorporate a more complex network structure
than a random regular network to analyze network structural eﬀects on intergenerational
inequality.

4

Numerical analysis with temporal networks

Social networks change when new friendships are created and existing ones are broken up.
If the change in social networks is governed by a speciﬁc process, each generation has a
speciﬁc feature depending on the time in the social network. The class of time-dependent10

1γ

networks is called temporal networks (Holme and Saramäki, 2019). We suppose that thechange in social networks over time follows a ﬁtness model (e.g. Kobayashi et al., 2019),
and calculate the quantum of beneﬁts that workers of a generation get, incremental tothose of the other generations, from their network structure.

4.1

Social network formation and degree distributions

Suppose that when entering the labor market (generation 0), a social network is createdsuch that the probability that each pair of workers has a friendship is given by ϕ ∈ (0, 1),
which is known as an Erdős-Rényi network. Then, f0 (k) is given by binomial distribution
such that
N −1 kf0 (k) =ϕ (1 − ϕ)N −k−1 .
(28)
kOnce a social network is created, the network changes following a ﬁtness model.
In a ﬁtness model, each worker has the activity parameter, and a pair of two workers
will connect by an edge with the probability depending on these activity parameters. If it
is allowed for the activity parameter to change, and it depends on the number of existing
friendships of each worker over time, the procedure in which workers make up friendships
is similar to preferential attachment processes (Barabási and Albert, 1999; Barrat andPastor-Satorras, 2005). Let aig be activity parameter of worker i in generation g andassume thatkig + 1
aig =
,
(29)
⟨kg ⟩ + 1
where ⟨kg ⟩ is the average number of friends of generation g.
Let Uig be the set of workers who are not friends of i in generation g. The probability
that worker i will connect with worker j ∈ Uig when they move to generation g + 1 isP rg (i, j) = λaig ajg ,(30)

where λ > 0 is a parameter. While new friendships are created in the network, the existingones are broken up with probability s ∈ (0, 1).
Based on Eqs. (28), (??), and (30), we obtain fg (k) for generation g > 0 by thefollowing simulation.
1. Generate Erdős-Rényi network with ϕ and set iterator l = 0.
2. Let A and B be the set of worker pairs having and not having a friendship, respectively.
3. Break up a friendship for each pair in A with probability s.
4. Create a new edge for each pair in B with probability P rg (i, j).
5. Increment l by one and if l < g, go back 2. Else, the simulation is ﬁnished, anddegree distribution fg (k) is computed.
We calibrate parameter set {s, λ, ϕ} to ﬁt the data in the next subsection.611

50
means from model
data
data fitting

number of friends

40

30

20

10

0

20

30

40
age

50

60

Fig. 4. Average number of friendships. Green plots describe the volume of the degree distributions
at ages 20, 30, 40, 50, and 60 produced by the ﬁtness model.

4.2

Parameterization

To obtain the numerical result, we calibrate parameters to ﬁt Japanese data. One period
is a quarter, and workers work from the 1st quarter of 20 to the 4th quarter of 64 years,
while they begin to search for jobs when they are in the last quarter of 19 years whichcorresponds to generation 0. There are 1000 workers for each generation, meaning thatN = 1000. We set discount factor β to 0.99. Home production b is 0.336 given by 60%
of average wage which is targeted at 0.56 (mentioned later). We give output yg by linear
function of generation g;2(1 − b)yg =g + b ∀g,ḡP
which assures us that yg ≥ b ∀g, the average output over generations ( ḡg=0 yg /(ḡ + 1))
equals to one, and workers’ productivity grows by age. The assumption of productivity
growth is adopted by Esteban-Pretel and Fujimoto (2014), in which worker productivity
grows stochastically for each individual. In our formulation, workers’ productivity grows
deterministically for simplicity. Other parameters are calibrated to ﬁt targets.
Calibration is divided into two steps. In the ﬁrst step, parameter set {s, λ, ϕ} iscalibrated. We compute the average number of friends by age based on International
Social Survey Programme (ISSP): Social Networks and Social Resources, 2017.7 In Fig. 4,
black dots indicate the averaged number of friends in each age and the orange broken lineis the ﬁrst order ﬁtting by OLS. To ﬁt the data, the parameter set is provided such that{s, λ, ϕ} = {3.21 × 10−5 , 0.0014, 0.023}, and in Fig. 4, the blue solid line is the average
number of friends calculated by the model. Also, we show the volume of the degree
distributions for ages 20, 30, 40, 50, and 60 based on the model by the green plots. With
the calibrated parameters, the model can roughly capture the average and dispersion of
the number of friends.
In the second step, based on the simulated degree distribution, the remaining parameters are calibrated to ﬁt 2017 Japanese data. The calibrated parameters are job opening
6

We actually use the distribution fg (k) averaged over 100 trials for each generation g tostochas∑exclude
˜gi (k)/100.
f
tic ﬂuctuation. In other words, letting f˜gi (k) be the realization value in trial i, fg (k) = 100
g=1
7
See calculation procedure in Appendix D.

12

Tab. 1. Parameters for numerical calculations.

Exogenous parameters
periodworking age
N number of workers in a generationyoutputbhome production
βdiscount factor

quarter
20-64
1000
2(1 − b)g/ḡ + b
0.336 (60% of avg. wage)
0.99

Calibrated parametersλscale of friendship making probabilitysprobability of breaking a friendshipϕprobability of friendships made at entering the market
cvacancy opening costηworker’s bargaining powerδ
job destruction rate
ψ eﬃciency in referral hiringγ
eﬃciency in the labor market

3.21 × 10−50.0014
0.023
6.310
0.024
0.022
0.355
15.37

cost c, worker’s bargaining power η, job destruction rate δ, eﬃciency in referral ψ, andeﬃciency in the labor market γ. We set the following ﬁve targets:
1. From Survey on Employment Trends, the ratio of referral matches to all matches is
0.213;P179
R
g=1 (1 − v)pg ug= 0.213.
P179
g=1 pg ug
2. From Penn World Table, the ratio of workers’ compensation to GDP is 0.56;
180X(1 − ug )wg

180

g=1

= 0.56,

3. and the labor market tightness, which is the ratio of the unemployment rate to thevacancy rate, is 1.5;v
= 1.5,uP
where u ≡ 180
g=1 ug /(180).
4. From Labour Force Survey, the unemployment rate among 20-24 year olds is 0.047
given by;20
X
ug
= 0.047,20
g=1

5. and the average unemployment rate among 20-64 year olds is 0.03;
180
X
ug
g=1

180

= 0.03.

Each of these targets determines c = 6.310, η = 0.024, δ = 0.022, ψ = 0.355, γ = 15.37
respectively. The resulting parameters are summarized in Tab. 1.
13

(a) wage
1.2(b) unemployment rate

model
data

model
data

0.045

1.1

0.040

1.0
0.035
0.9
0.030

0.8
0.7

0.025
20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64

20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64

age classFig. 5. Wages and unemployment rates at the equilibrium. Wages are normalized such that the
average is equal to one in (a).

4.3

Numerical solution and social network premium

The results in Fig. 5 show the average by ﬁve-year age group, in which blue and orange lines
are the values by the model and by data, respectively. In Fig. 5a, we normalize the average
wage to be one to compare results from the model and data. The model mainly succeeds
in capturing age-dependent tendencies in the context of wages and unemployment rates.
Then, how much does each generation beneﬁt from the social network in the equilibrium?
Workers can receive additional wages from their social network since they can lower
unemployment probabilities through referral hiring. We deﬁne the social network premium
as the diﬀerence in the equilibrium values achieved with and without referrals. Here we
calculate the average wage and unemployment premium for each generation.
Workers in a social group beneﬁt not only from the mean number of friends in the
network, but also from the structure of the network (Ogisu, 2022; Tassier and Menczer,
2008). We can also compute the equilibrium values if the structure of networks does
not change over time, and use it for calculating how much workers beneﬁt from network
structures.
Let wg,tem , wg,stat , and wg,no be an equilibrium wage of generation g when all workers
can use referrals and social networks change over time, when all workers can use referrals
but social networks do not change over time, and when all workers cannot use referrals,
respectively. We denote a social network premium to the wage of generation g by ρw
g , anddeﬁne it asρw(31)g ≡ wg,tem − wg,no .
It can be decomposed such thatρw
g = wg,tem − wg,stat + wg,stat − wg,no .
|
{z
} |
{z
}
structural eﬀect(32)

common eﬀect

When social network structures do not change, all generations have the same network
structure and degree distribution following Eq. (28). Thus, the additional premium (orpenalty) from time-dependent structures of social networks can be captured by the difference between wg,tem and wg,stat , and we call it structural eﬀect and the rest common
eﬀect.
In the same manner, we also deﬁne social network premium to the unemployment rateof generation g as
ρug ≡ ug,no − ug,tem ,14

(a) wage premium(b) unepmloyment premium
0.00200.015
0.00150.010

premiumcommon effectstructural effect

0.00100.005

0.00050.000

0.000020-2425-2930-3435-3940-4445-4950-5455-5960-64

20-2425-2930-3435-3940-4445-4950-5455-5960-64

age

Fig. 6. Premiums from social networks. Social network premiums are positive while parts of thepremiums are counteracted by structural eﬀects for most generations.

and decompose such thatρug = ug,no − ug,stat + uu,stat − uu,tem .
{z
} |
{z
}
|
common eﬀect(33)

structural eﬀect

Note that the sign of ρug is reversed in the deﬁnition of ρw
g since the premium in unemployment rates is captured as the decrease.
The calculated premiums averaged by the ﬁve-year age group are shown in Fig. 6. All
generations receive premiums from their social networks and they are up to 1.6%pt atage class 55-59 in wages and up to 0.2%pt at the age class 25-29 in unemployment rates.
The premium is the lowest in the youngest group. In turn, the youngest group can geta relatively large structural eﬀect, meaning that workers in the group have good network
structures in referral hiring. This can be interpreted as young workers being unable to use
referrals frequently even if they have social networks since most friends are unemployed.
Thus, when workers use referrals, most job slots are ﬁlled by middle- or old-age workers.
The relationship between wage and unemployment premiums is not monotonic. In
particular, the wage premium of age class 60-64 decreases from that of the 55-59 classalthough the unemployment premium does not decrease as much. This is because workers
in age 60-64 have little room for bargaining compared to young ones since they will retiresoon. Workers belonging to the last quarter of age 64, for example, cannot get additional
premiums whenever the change of friendships happens in the other generation because
they receive wḡ or b and exit the market. In other words, their incomes are ﬁxed by theterminal conditions.
While there are positive premiums overall, structural eﬀects of premiums decreaseas age classes move up, which is caused by the congestion within the network for eachgeneration. The penalties from the network structures are up to 0.3%pt at age class 55-59
in wages and up to 0.03%pt at age class 60-64 in unemployment rates. Based on Eq. (30),
new friendships are created proportional to the number of friends of workers; in otherwords, a worker with more friends can get new friends with a higher probability than aworker with fewer friends. By this mechanism, workers in the social network are divided
into two groups as time goes on: some workers who are popular and lots of workers who do
not have many friends. A worker group with such a network structure is at a disadvantage
in the labor market (Ogisu, 2022) since vacancy information concentrates in the hands ofsome popular workers, and as a result, older generations incur penalties from their network
structure more than younger ones.
15×10−5(a) differences in wage premiums(b) differences in unemployment rate premiums

×10−6

6

6

4

4

2
2
0
0
−2

20-2425-2930-3435-3940-4445-4950-5455-5960-64

20-2425-2930-3435-3940-4445-4950-5455-5960-64

age classFig. 7. Spillover eﬀects by increment in the number of friends at the 1st quarter of age 40. Bars
show changes in the structural eﬀects by deﬁnition. The class including age 40 obviously
beneﬁts although the others do not. In the classes more than age 45, the eﬀects are relatively
small since they are partly oﬀset by the shift of Beveridge curves.

4.4

An example of spillover eﬀects

In this subsection, we provide an example of spillover eﬀects by an increment in the numberof friends of a generation. We choose the 1st quarter of age 40 (g = 81) for the generations
in which degree distribution will change. To describe the increment in the number offriends, the mass of the k is moved to the k + 2 for all k in the degree distribution. This
shift in the degree distribution changes the average number of friends from 20.1 to 22.1,
which is an increase of about 10% only for the generation.8
We compute the new equilibrium and record diﬀerences in old and new ones for eachage class, shown in Fig. 7. All of these changes in premiums are explained by the change
in structural eﬀect by deﬁnition. Although the networks are complex compared to thecase in Sec. 3, the basic mechanism is generally the same. The age class 40-44 years,
including the generation in which the degree distribution shifts right, obviously obtains
the additional premiums to unemployment rates, meaning that the average unemployment
rate decrease. However, premiums decrease for the older age classes compared with age
45, therefore, these unemployment rates at the new equilibrium increase. These changes
are consistent with the case in Fig. 3b and 3c. Equilibrium wages decrease except for theage class 40-44 while the absolute values of the changes are smaller in the older ages than
the younger ages since shifts of the Beveridge curves in the older generations can mitigate
the decrease in these unemployment rates.
In this section, we demonstrate that workers beneﬁt from their social networks while
the beneﬁt can be partly counteracted by the congestion led by these network structures.
In the calibrated case suited to the Japanese context, spillover eﬀects can be negative
because of a decrease in the vacancy rate.

5

Conclusion

In this paper, we demonstrate the generation-dependent premiums from their social networks in a search and matching model with overlapping generations. These premiums areshown to be small for young workers compared to middle- and old-aged ones. A crucial
reason is that young workers cannot utilize referrals suﬃciently even if they have many8

We calculate two other examples in Appendix E; increase in connectivity of young and old generation.

16

friends since many of them are also unemployed. This implies that beneﬁts from social
networks vary in response to the unemployment dynamics. We note here that premiums
are partially oﬀset by congestion caused by network structures for most generations.
A change in the connectivity of a network aﬀects not only the generation of which thedegree distribution changes, but also the other generations through the changes in thevacancy rate in equilibrium. When the vacancy rate increases in response to a change inthe degree distribution, then the unemployment rates of the other generations decrease.
Even if, however, the vacancy rate decreases, it is possible that the unemployment ratesof older generations become lower than before, or the increments in unemployment rates
are partially oﬀset due to the shift of the Beveridge curves.
Our model can be extended in various directions. Among them, the following three areimportant and hence worth investigating. First, an alternative model could be presented
to be used for a realistic description of social networks. We have used a ﬁtness model withsimple rules to describe the dynamics in social networks. However, other speciﬁcations
than this one are also possible. We use only the average number of friendships from data
for the calibration target. If a data set of detailed social networks in the real economy
becomes available, the model speciﬁcation will be modiﬁed in a way to capture the actual
characteristics of networks (e.g. clustering coeﬃcients, diameters).
Second, social networks could be endogenized. In the model presented in this paper,
workers do not invest in keeping or making friendships. If workers expect to receive referrals from their friends, they have incentives to make friends (e.g. Galenianos, 2021; Galeotti
and Merlino, 2014; Horvath and Zhang, 2018). Including the endogenized network, thedynamics of variables in equilibrium (e.g. unemployment rates, job arrival rates) are determined as a result of the interaction between the social network dynamics and workers’
decisions, which allows us to analyze premiums from a temporal network in a realistic
sense.
Third, on-the-job search could be included. Referrals can work signiﬁcantly to step
workers’ careers in reality. One of the natural extensions is to make job productivity
stochastic. As a result, the referral process becomes complicated because all workers wait
for good job information. Since a job passed down from a friend is not better than thefriend’s, if workers hope to get better-paid jobs, they need to wait for such a job in thelabor market which is a tiny probability, or to form friendships with workers who earnmuch. Thus, in a model with on-the-job search, positions on the social networks, i.e. whoyou know, will become more crucial to determine premiums than in the model presented
in this paper.
An important remark exists in interpreting the results put forward in the paper. To
introduce network structure into the model, we employ degree distributions for the information on the network structures, similar to Ioannides and Soetevent (2006) and Ogisu(2022). This formulation is unable to capture the micro characteristics of the network
structures like clustering coeﬃcients. Thus, social networks in this paper cannot be distinguished from conﬁguration models. In a conﬁguration model, workers’ degree distribution
is prespeciﬁed, while the connection between workers is given at uniformly random, andhence clustering coeﬃcients are equal to zero with a suﬃciently large number of nodes(Catanzaro et al., 2005; Newman, 2018), although in an assortative mixing model, it ispossible that clustering coeﬃcients are positive (Jackson and Rogers, 2007). The impactof micro characteristics of network structure is one of the most important issues to be
examined in the future.

17

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20

Appendix
A

Proof of Proposition 1

We respectively rewrite Sg , pg togY

Sg = (y − b) 1 +

∆τ +

τ =g

∆τ + · · · +τ =gḡ−1 YmX= (y − b) 1 +

g+1
Y

!
∆τ

τ =g

!

∆τ

ḡ−1
Y

,

m=g τ =g

Pg = ψγv(1 − ug )1 − (1 − ug )k̄g
k̄g ugk̄g −1=ψγv(1 − ug ) X(1 − ug )κ ,k̄g
κ=0

pg = γv + (1 − γv)[1 − (1 − Pg )k̄g ]= γv (1 − Pg )k̄g + 1 − (1 − Pg )k̄g
k̄g −1k̄g= γv (1 − Pg )

+ Pg

X

(1 − Pg )κ ,

κ=0

where ∆g ≡ β(1 − δ − pg η). Therefore, we can calculatek̄g −1k̄g −1
X
X
u
)
pg
ψγ(1
−
g 
= γ(1 − Pg )k̄g +(1 − ug )κ  (1 − Pg )κ  .
v
k̄g
κ=0
κ=0

(34)

When v approaches zero, for all g ∈ {0, 1, · · · ḡ}lim Pg = 0,

lim pg = 0, lim ug = 1,v→0
1 − [β(1 − δ)]ḡ−g+1≡ Sg,
lim Sg = (y − b)v→0
1 − β(1 − δ)
pg
lim
= γ.
v→0 v
v→0

v→0

Thus,
lim

v→0

ḡ
X
g=0

X
pg
ug Sg = γ
Sg.
v
ḡ

g=0

As v → 1/γ, for all g ∈ {1, 2, · · · , ḡ}lim Pg = ψ,v→ γ1

lim pg = 1,v→ γ1

lim Sg = (y − b)v→ γ1

lim ug = 0,v→ γ1

1 − [β(1 − δ − η)]ḡ−g+1≡ Sg,
1 − β(1 − δ − η)pg
= γ,v→ γ1 v
lim

21

(35)

and for g = 0 with u0 = 1,
lim P0 = 0,

lim p0 = 1,

v→ γ1

v→ γ1

lim S0 = (y − b)

v→ γ1

lim

v→ γ1

1 − [β(1 − δ − η)]ḡ+1≡ S0,
1 − β(1 − δ − η)

p0
= γ.
v

Thereby,k̄g −1

lim

v→ γ1

X

ug

g=0

pg
Sg = γS 0 .
v(36)

As a result,Xc(ḡ + 1)
γS 0 <<γSgβ(1 − η)
ḡ

g=0

⇐⇒


ḡ 
γβ(1 − η)(y − b) X 1 − [β(1 − δ)]ḡ−g+1
γβ(1 − η)(y − b) 1 − [β(1 − δ − η)]ḡ−g+1<1<c(ḡ + 1)
1 − β(1 − δ − η)c(ḡ + 1)
1 − β(1 − δ)
g=0

is the suﬃcient condition for the existence of the equilibrium.

B

Proof of Lemma 1
∂Pg
⋛ 0,
∂ug
⇐⇒ (k̄g ug + 1)(1 − ug )k̄g ⋛ 1.

(37)

LHS of (37) decreases in ug and approaches one when ug → 0. Therefore,
∂Pg
≤ 0.
∂ug
And we can get
∂pR
∂Pg
g
= k̄g (1 − Pg )k̄g −1
≤0
∂ug
∂ug
∂pg
≤ 0.
∂ug

∀ug ∈ (0, 1],

By deﬁnition,
P0 = 0,p0 = γv,

and




 ∂u 
∂Pg
∂ug 
ψγ
g
k̄g +1k̄g
ug − v=
1 − ug − (1 − ug )+ vug (k̄g + 1)(1 − ug ) − 1∂v∂v∂v
k̄g u2g
 


∂ug 
ψγ
k̄g +1k̄gk̄g +1=
ug 1 − ug − (1 − ug )+v
ug (k̄g + 1)(1 − ug ) + (1 − ug )−1∂v
k̄g u2g
i ∂u
Pg
ψγv h
g
k̄g
k̄g +1
=
+
(
k̄
+
1)u
(1
−
u
)
+
(1
−
u
)
−
1
g
g
g
g
2
v
∂v
k̄g ug
h
i
Pg
∂ug
ψγv
=
(k̄g ug + 1)(1 − ug )k̄g − 1
+
,
2
v
∂v
k̄g ug22

where (k̄g ug + 1)(1 − ug )k̄g − 1 < 0 ∀ug ∈ (0, 1] and


∂pg
∂Pg
= γ(1 − Pg ) + k̄g (1 − γv)(1 − Pg )k̄g −1
∂v
∂v
∂Pg
= γ(1 − Pg )k̄g + k̄g (1 − γv)(1 − Pg )k̄g −1
"∂v
#
h
i ∂u
P
ψγv
g
g
= γ(1 − Pg )k̄g + k̄g (1 − γv)(1 − Pg )k̄g −1+(k̄g ug + 1)(1 − ug )k̄g − 1v∂v
k̄g u2g
Pg= γ(1 − Pg )k̄g + k̄g (1 − γv)(1 − Pg )k̄g −1
v
h
i ∂u
g
k̄g −1 ψγv
k̄g+ (1 − γv)(1 − Pg )
(
k̄
u
+
1)(1
−
u
)
−
1
g
g
g
u2g
∂v


Pg
(1 − Pg )k̄g −1= γ(1 − Pg ) + k̄g (1 − γv)
v
i ∂u
ψγv h
g
+ (1 − γv)(1 − Pg )k̄g −1 2 (k̄g ug + 1)(1 − ug )k̄g − 1
ug
∂v


Pg
= γ(1 − Pg ) + k̄g (1 − γv)(1 − Pg )k̄g −1
v
i ∂u
ψγv h
g
+ (1 − γv)(1 − Pg )k̄g −1 2 (k̄g ug + 1)(1 − ug )k̄g − 1
ug
∂v
"
#
1 − ug − (1 − ug )k̄g +1= γ(1 − Pg ) + (1 − γv)ψγ(1 − Pg )k̄g −1
ug
i ∂u
ψγv h
g
+ (1 − γv)(1 − Pg )k̄g −1 2 (k̄g ug + 1)(1 − ug )k̄g − 1,ug∂v
and by the law of motion,
∂ug
∂pg−1
∂ug−1
=−
ug−1 + (1 − δ − pg−1 )
∂v
∂v 
∂v

Pg−1
= −ug−1 γ(1 − Pg−1 ) + k̄g (1 − γv)(1 − Pg−1 )k̄g −1
v
i ∂u
ψγv h
g−1
− ug−1 (1 − γv)(1 − Pg−1 )k̄g −1 2(k̄g ug−1 + 1)(1 − ug−1 )k̄g − 1
∂v
ug−1
∂ug−1
+ (1 − δ − pg−1 )
∂v
∂u2
lim
= −γ − γ(1 − δ)
v→0 v
thus,
∂u0
= 0,
∂v

∂u1
= −γ < 0,∂v

and for g = 2,
∂u2
< 0,
∂v
since
∂p1
> 0.
∂v

∂P1
> 0,
∂v
23

And after that, similarly, for all g ≥ 2,
∂Pg
> 0,
∂v

∂ug
< 0,
∂v

∂pg
> 0.
∂v

Consequently,
"
( g
)
g+1
ḡ−1
Y
Y
Y
∂pg 1
∂Sg
= −βη(y − b)∆τ +∆τ + · · · +∆τ∂v∂v ∆g τ =g
τ =g
τ =g
(g+1
)
ḡ−1
Y
Y
∂pg+1 1∆g+1 + · · · +∆τ+∂v ∆g+1 τ =g
τ =g
..
.
∂pḡ−2 1+∂v ∆ḡ−2∂pḡ−1 1+∂v ∆ḡ−1

(ḡ−2
Y

∆τ +

τ =g

(ḡ−1
Y

ḡ−1
Y

)
∆τ

τ =g

)#
∆ḡ−1

<0

∀g ∈ {0, 1, · · · , ḡ − 1},

τ =g

∂Sḡ
= 0.
∂v
Diﬀerentiating G(v) by v,∂(ug (pg /v)Sg )∂v∂ug pg∂(pg /v)
pg ∂Sg=Sg + ug
Sg + ug
∂v v
∂v
v ∂v
∂ug pgpg ∂Sg=Sg + ug
∂v v
v ∂v

g −1
 k̄X
ψγ ∂ug k̄g −1 ∂Pg
k̄g+ ug Sg −γ k̄g (1 − Pg )(1 − Pg )κ
+(k̄g + 1)(1 − ug ) − 1∂v∂v
k̄g ug

κ=0

X
∂Pg
ψγ 
1 − ug − (1 − ug )k̄g +1
κ(1 − Pg )κ−1∂v
k̄g ug
κ=1

g −1
 k̄X
ψγ ∂ug k̄g +1−
1 − ug − (1 − ug )(1 − Pg )κ 
2∂v
k̄g ugκ=0k̄g −1−=

pg ∂Sg∂ug pgSg + ug
∂v v
v ∂v


g −1

 k̄X
∂P
∂P
ψγ
g
g
+ ug Sg −γ k̄g (1 − Pg )k̄g −1
1 − ug − (1 − ug )k̄g +1−
κ(1 − Pg )κ−1∂v∂v
k̄g ugκ=1X
ψγ ∂ug + Sg(1 − Pg )κ
(k̄g + 1)(1 − ug )k̄g − 1k̄g ∂v
κ=0k̄g −1X
ψγ ∂ug − Sg
1 − ug − (1 − ug )k̄g +1(1 − Pg )κ
k̄g ug ∂v
κ=0k̄g −124

= ug

pg ∂Sg
v 
∂v

= ug

pg ∂Sg
v ∂v



g −1

 k̄X
∂P
∂P
ψγ
g
g
− ug Sg γ k̄g (1 − Pg )k̄g −1+
1 − ug − (1 − ug )k̄g +1
κ(1 − Pg )κ−1∂v∂v
k̄g ugκ=1k̄g −1


X
∂ug 
ψγ
+ Sg
γ(1 − Pg )k̄g +(k̄g + 1)(1 − ug )k̄g − 1(1 − Pg )κ 
∂v
k̄g
κ=0

k̄g −1X∂Pgψ
1 − ug − (1 − ug )k̄g +1
κ(1 − Pg )κ−1 − ug Sg γ k̄g (1 − Pg )k̄g −1 +∂v
k̄g ug
κ=1


g −1

 k̄X
∂ug 
ψ
γ (1 − Pg )k̄g ++ Sg(k̄g + 1)(1 − ug )k̄g − 1(1 − Pg )κ  ,
∂v
k̄g
κ=0

where only the last term can be positive. When the bracket in the last term becomes
positive, all terms in the equation become negative since ∂ug /∂v < 0. For g = 0, thebracket in the last term is
1 − ψ > 0,
and for g ∈ {1, 2, · · · , ḡ}, it becomes such that


g −1

 k̄X
ψ
(k̄g + 1)(1 − ug )k̄g − 1(1 − Pg )κ  = 1 > 0,lim (1 − Pg )k̄g +
1
k̄
v→ γ
g
κ=0
Thus, for each g ∈ {0, 1, · · · ḡ} there exists v g ∈ (0, 1/γ) such that for v ∈ (v g , 1/γ)
∂ug (pg /v)Sg< 0.
∂v
Therefore, letv ≡ max{vg }ḡg=0 ,then for v < v, ∂ug (pg /v)Sg /∂v < 0 for all g, which is suﬃcient for ∂G(v)/∂v < 0.

C

Proof of Proposition 3

For k̄g > 0,iγψv(1 − ug ) h∂Pg=
1 − k̄g log(1 − ug ) (1 − ug )k̄g − 1 < 0,∂ k̄g
ug k̄g∂pgk̄g ∂Pg= −(1 − γv)(1 − Pg )k̄g log(1 − Pg ) −1 − Pg ∂ k̄g
∂ k̄g


k̄g ∂Pg− log(1 − Pg )= (1 − γv)(1 − Pg )k̄g
1 − Pg ∂ k̄giγψv(1 − ug ) hk̄gk̄g
= (1 − γv)(1 − Pg )1 − k̄g log(1 − ug ) (1 − ug ) − 1 − log(1 − Pg ) .
(1 − Pg )ug
25

log(1 − Pg ) increases monotonically in k̄g ,γψv(1 − ug )lim log(1 − Pg ) = log 1 +log(1 − ug ) < 0,ug
k̄g →0lim log(1 − Pg ) = 0,k̄g →∞k̄g ∂Pg= 0.
1−
Pg ∂ k̄g
k̄g →0lim

Therefore, there is k̄g = k̂g such that ∂pg /∂ k̄g = 0, at which pg is maximized, and∂pg⋛ 0 ⇐⇒ k̄g ⋚ k̂g∂ k̄g
Intergenerational propagations are such that∂Pg
∂Pg ∂ug=∂kg−1∂ug ∂kg−1∂Pg ∂pg−1= −ug−1∂ug ∂ k̄g−1∴∂Pg⋛ 0 ⇐⇒ k̄g−1 ⋚ k̂g−1
∂kg−1

and
∂Pg
∂pg
= k̄g (1 − γv)(1 − Pg )k̄g −1.
∂ k̄g−1∂ k̄g−1
Similarly,∂Pg+m∂Pg+m ∂ug+m=,∂ug+m ∂ k̄g
∂ k̄g
∂Pg+m
∂pg+m
= k̄g+m (1 − γv)(1 − Pg )k̄g+m −1
∂ k̄g
∂ k̄g
∂Pg+m ∂ug+m= k̄g+m (1 − γv)(1 − Pg )k̄g+m −1∂ug+m ∂ k̄g


∂ug+m
∂pg+m−1∂pg+m−2∂ug+m−2=−
ug+m−1 + (1 − δ − pg+m−1 ) −ug+m−2 + (1 − δ − pg+m−2 )
∂ k̄g
∂ k̄g
∂ k̄g
∂ k̄g
∂pg+m−1∂pg+m−2=−
ug+m−1 − (1 − δ − pg+m−1 )
ug+m−2∂ k̄g∂ k̄g∂ug+m−2+ (1 − δ − pg+m−1 )(1 − δ − pg+m−2 )
∂ k̄g
g+m−1
g+m−1
Y
Y
∂pg
∂pg+m−1∂pg+m−2(1 − δ − pτ ) − · · · −(1 − δ − pτ )ug+m−1 −ug+m−2
ug
∂ k̄g
∂ k̄g
∂
k̄
g
τ =g+m−1
τ =g+1


g+m−1
g+m−1
X ∂pr−1Y∂pg+m−1= −
(1 − δ − pτ )
ug+m−1 +
ur−1
∂ k̄g
∂
k̄
g
τ =r
r=g+1

=−

26

And for m ≥ 0,
∂Sg+m
∂pg+m 1= −βη(y − b){∆g+m + ∆g+m ∆g+m+1 + · · · + ∆g+m · · · ∆ḡ−1 }∂ k̄g∂ k̄g ∆g+m
∂pg+m+1
1+{∆g+m ∆g+m+1 + · · · + ∆g+m · · · ∆ḡ−1 }∂ k̄g ∆g+m+1..
.
∂pḡ−2 1{∆g+m · · · ∆ḡ−2 + ∆g+m · · · ∆ḡ−1 }+∂ k̄g ∆ḡ−2∂pḡ−1 1+{∆g+m · · · ∆ḡ−1 }∂ k̄g ∆ḡ−1
"
( g+m
)
g+m+1
ḡ−1
Y
Y
Y
∂pg+m 1= −βη(y − b)∆τ +∆τ + · · · +∆τ∂ k̄g ∆g+m τ =g+m
τ =g+mτ =g+m(g+m+1)ḡ−1YY∂pg+m+1
1+∆τ + · · · +∆τ∂ k̄g ∆g+m+1 τ =g+m
τ =g+m..
.
∂pḡ−2 1+∂ k̄g ∆ḡ−2( ḡ−2Y∆τ +τ =g+m

ḡ−1
Y

)
∆τ

τ =g+m( ḡ−1)#Y∂pḡ−1 1∆τ+∂ k̄g ∆ḡ−1 τ =g+m"g+m
1 ∂pg+m Y= −βη(y − b)∆τ∆g+m ∂ k̄g τ =g+m


+

+

∂pg+m
∂pg+m+1
1+∆g+m ∂ k̄g∆g+m+1 ∂ k̄g1

 g+m+1Y∆ττ =g+m∂pg+m∂pg+m+1∂pg+m+2
1
1++∆g+m ∂ k̄g∆g+m+1 ∂ k̄g∆g+m+2 ∂ k̄g
1

..
.
ḡ−1
X

1 ∂pτ
+
∆
τ ∂ k̄gτ =g+m

!

#

ḡ−1
Y

∆τ .

τ =g+mThus, since ∂Pg /∂ug < 0,k̄g ⋚ k̂g
∂pg
=⇒
⋛0
∂ k̄g
∂ug+1
=⇒
⋚0
∂ k̄g
∂pg+1
=⇒
⋛0
∂ k̄g
27

 g+m+2Yτ =g+m

∆τ

..
.
∂ug+m
⋚0
∂ k̄g
∂pg+m
=⇒
⋛0
∂ k̄g
∂Sg+m
=⇒
⋚ 0.
∂ k̄g
=⇒

D

Construction of the number of friendships

We use International Social Survey Programme (ISSP): Social Networks and Social Resources.9 In the questionnaire, a respondent chooses how many people she has contact
with on a typical weekday from options that include “0-4,” “5-9,” “10-19,” “20-49,” “5099,” “100 or more,” “Can’t choose,” and “No answer.” We exclude samples answering thelast two, and set the middle value of each range to a representative value(e.g. 2 for range“0-4”). We set it 100 if the answer is “100 or more.” Lastly, we take the average of thevalue by each age classes and show it in Fig. 4.

E

Spillover eﬀects from young and old age

We check here the spillover eﬀects from other generations than middle age not discussed
in Sec. 4.4. Fig. 8 and 9 are respectively spillover eﬀects by increasing in connectivity inthe social network of 1st quarter of age 20 and of 3rd quarter of age 64. In both cases,
the network connectivity increase by about 10% by the same procedure in Sec. 4.4.
In Fig. 8, young workers incur relatively large penalties despite the increase in connectivity. When the network connectivity expands in the young generation, Beveridge curvesof middle and old generations also shift. Thus, the probability of referrals increases for
all generations. It leads to additional job-worker matches in many generations, and toa decrease in the equilibrium vacancy rate. However, young generations have still highunemployment rates and cannot use referrals well. As a result, the beneﬁts do not go toyoung generations, but to middle and old generations.
In Fig. 9, by contrast, all generations can beneﬁt from the increase in network connectivity. In this case, the Beveridge curve of the 3rd quarter of age 64 shifts lower althoughthese curves of other generations do not. A part of new entrant jobs allocates to thegeneration in which network connectivity increase by referrals while the other part goes tothe labor market. It makes the tightness of the labor market loose and improves the job
arrival rate in the labor market. Consequently, all generations beneﬁt from the change.

9

https://search.gesis.org/research_data/ZA6980, (access in July 2022)28

×10−7(a) differences in wage premiums(b) differences in unemployment rate premiums

0.00

0.0

×10−6

−0.25
−0.50

−0.5

−0.75

−1.0

−1.00
−1.25

−1.5

−1.50

−2.0
20-2425-2930-3435-3940-4445-4950-5455-5960-64

20-2425-2930-3435-3940-4445-4950-5455-5960-64

age classFig. 8. Spillover eﬀects by increment in the number of friends at the 1st quarter of age 20. Bars
show changes in the structural eﬀects.

5

×10−6

(a) differences in wage premiums(b) differences in unemployment rate premiums

×10−6
6

4

5
4

3

3

2

2
1
0

1
0

20-2425-2930-3435-3940-4445-4950-5455-5960-64

20-2425-2930-3435-3940-4445-4950-5455-5960-64

age classFig. 9. Spillover eﬀects by increment in the number of friends at the 3rd quarter of age 64. Bars
show changes in the structural eﬀects.

29