This paper develops a model where two agents in different sectors face uncorrelated income risks and mutually self-insure. We discuss how the rent arising from risk pooling modifies the wage distribution in the sector where the employer behaves as a monopsonist.risk pooling, family transfers

St-Amour, Pascal

Vencatachellum, Désiré

English

Research Papers in Economics

Inter-Sectorial Risk Pooling and Wage Distributions
￿
Pascal St-Amour
y
Université Laval and CRÉFA
Désiré Vencatachellum
Hautes Études Commerciales, Montréal.
August 26, 1998
Abstract
This paper develops a model where two agents in different sectors face uncorrelated income risks and mutually
self-insure. We discuss how the rent arising from risk pooling modiﬁes the wage distribution in the sector where the
employer behaves as a monopsonist.
Keywords: risk pooling, family transfers.
JEL classiﬁcation: O130, O150, O170.
￿We have beneﬁted from discussions with Sylvain Dessy, Bernard Fortin and Simon Van Norden. We are responsible for any remaining errors
and omissions.
yCorresponding author: Département d’Économique, Université Laval, Cité Universitaire, Québec, Canada, G1K-7P4, e-mail: pascal.st-
amour@ecn.ulaval.ca, tel.: (418) 656-5610, fax: (418) 656-7798.1 Introduction
Inter-sectorial transfers by members of extended families are an important component of household incomes in most
developing countries (World Bank, 1994). These transfers arise in part because of incomplete insurance markets;
agents who cannot hedge against crop failure, health problems or unemploymentrisks use non-marketmechanisms as
a substitute (Ligon et al., 1997). Typically extended families have a comparative advantage in providing such services
because of superior information that mitigate moral hazard and adverse selection problems (Pollak, 1985). Moreover,
most of the labour force in those economies is employed in agriculture (Larson and Mundlak, 1997, Table 2). Land,
an essential input in that sector, is controlled by a small number of landlords (Tomich et al., 1995) who often use a
collusive strategy when hiring workers (Bardhan, 1989).
This note focuses on the link between inter-sectorial transfers between agents with symmetric information, and
wage distributions in labour markets characterized by intra-sectorial collusion among employers. We ﬁrst illustrate
how such transfers modify the distribution of wages paid by an employer who behaves as a monopsonist. Secondly,
we indicate why transferscan imply a transmission mechanism for wage distributionsacross sectors. Finally, we show
why, in the absence of inter-sectorial collusion, the economy may be stuck in a poverty trap.
2M o d e l
Consider an economy with two sectors denoted
i
=
1
;
2, each with many identical agents (workers) and a small
number of employers. Agents do not have access to a storage technology, while they derive utility from consump-
tion exclusively, and are expected-utility maximizers with monotone and concave Von Neumann-Morgenstern utility
function
u
(
￿
).
Let
w
i
2
f
w
i
;
w
i
g,w h e r e
w
i
>
w
i , denote the stochastic income of a representative agent employed in sector
i,
with
p
i
￿
P
r
(
w
i
=
w
i
). Inter-sectorialincomesareindependentwhileintra-sectorialincomesare perfectlycorrelated.
We abstract from migration across sectors. The realization of the agents’ income is common knowledge to agents, but
notto employers. Thisstructureprecludesanyformof intra-sectorialrisk pooling,includingself-insuranceandmarket
insurance. Consequently,two risk-averse agents may agree to pool risk across sectors and transfer part of their income
1to each other (Kocherlakota, 1996). Let
t
i
>
0 denote the transfer from a representative agent in sector
i (henceforth
agent
i) to an agent in the other sector. This transfer is determined after some bilateral bargaining process – which is
taken as given in this analysis – and takes place when only one of the two agents receives the low income. In this case,
the high-income earner transfers part of his revenues to the low-income one.1
Agreement to the transfer scheme by agent 1 implies that his expected utility is greater under risk pooling than
under autarky, i.e.:
r
1
￿
p
1
p
2
u
(
w
1
)
+
p
1
(
1
￿
p
2
)
u
(
w
1
￿
t
1
)
+
(
1
￿
p
1
)
p
2
u
(
w
1
+
t
2
)
+
(
1
￿
p
1
)
(
1
￿
p
2
)
u
(
w
1
)
￿
[
p
1
u
(
w
1
)
+
(
1
￿
p
1
)
u
(
w
1
)
]
￿
0
:
(1)
where
r
1 is agent 1’s surplus. After some simpliﬁcation, inequality (1) can be rewritten as:
p
2
￿
p
1
[
u
(
w
1
)
￿
u
(
w
1
￿
t
1
)
]
p
1
[
u
(
w
1
)
￿
u
(
w
1
￿
t
1
)
]
+
(
1
￿
p
1
)
[
u
(
w
1
+
t
2
)
￿
u
(
w
1
)
]
￿
p
￿
2
; (2)
where
p
￿
2 is the lowest probabilitythat agent 2 receives
w
2 such that agent 1 acceptsthe risk-poolingagreement. Using
the equivalent of inequality (2) for agent 2, we obtain:
p
1
￿
p
2
[
u
(
w
2
)
￿
u
(
w
2
￿
t
2
)
]
p
2
[
u
(
w
2
)
￿
u
(
w
2
￿
t
2
)
]
+
(
1
￿
p
2
)
[
u
(
w
2
+
t
1
)
￿
u
(
w
2
)
]
￿
p
￿
1
: (3)
It can be shown that a necessary condition for transfers to take place between agents is that the marginal beneﬁt
from the transfer received must be at least as large as the marginal loss from the transfer paid:
u
(
w
i
+
t
j
)
￿
u
(
w
i
)
￿
u
(
w
i
)
￿
u
(
w
i
￿
t
i
)
; (4)
1One example of such a transfer scheme is when
w
1
=
w
2
=
w and
w
1
=
w
2
=
w and both agents have the same bargaining power.
Equalizing marginal rates of substitution yields transfers given by the unconditional mean income, i.e.
w
￿
t
i
=
w
+
t
i which simpliﬁes to
t
i
=
1
=
2
(
w
+
w
). Other elements that determine the transfers
t
i might include non-insurance motives such as altruism, over which preferences
are not deﬁned in our model.
2for
i
;
j
=
1
;
2and
i
6
=
j. Under condition(4), the inequalities (2) and (3)deﬁne a convexcontractset in the probability
space, as illustrated in the left panel of Figure 1. The transfer scheme occurs only if
(
p
1
;
p
2
)lie in the shaded region.2
Demand for labour in sector 1 is characterized by imperfect competition with collusion among employers. As in
Bencivengaand Smith (1997),a representativeprincipaluses a mixed strategy and announcesan employmentcontract
which consists of the probability
p
1 that agent 1 will obtain
w
1, taking
p
2 and all wages as given.3 For now, sector 2
is treated as exogenous in this analysis. The principal chooses
p
1 to maximize agent 1’s risk-pooling surplus
r
1.H e
does so because, behaving as a monopsonist, he will eventually be able to extract this rent from agent 1 through some
mechanism left implicit in our analysis. Monotonicity of
u
(
￿
) in equation (1) implies that
r
1 is decreasing in
p
1:
@
r
1
@
p
1
=
(
p
2
￿
1
)
[
u
(
w
1
)
￿
u
(
w
1
￿
t
1
)
]
￿
p
2
[
u
(
w
1
+
t
2
)
￿
u
(
w
1
)
]
<
0
:
Consequently, the principal chooses the lowest
p
1 which satisﬁes the participation constraint for agent 2 given by
inequality (3), and sets
p
1
=
p
￿
1 (see the left panel of Figure 1).
Following an exogenousincrease in
p
2, the relative risk of agent 2 of receiving
w
2 decreases compared to the risk
faced by agent 1 of obtaining the low wage. Hence, a risk-pooling contract becomes less attractive to agent 2. The
principal who wishes to maintain the risk-sharing arrangement between the two agents must also raise the probability
of the high outcome for agent 1. Therefore, risk-sharing between agents in different sectors allows for an alternative
inter-sectorial transmission mechanism for wage distributions.
Moreover, under perfect intra-sectorial collusion but in the absence of inter-sectorial collusion, extracting the rent
created by risk pooling may lead to a poverty trap. In the right panel of Figure 1, if the principal in sector 2 is also a
monopsonistic Stackleberg player, its strategies are symmetric to those of sector 1: this employer sets
p
2
=
p
￿
2, taking
p
￿
1 as given. Subsequently, the principal in sector 1 responds by choosing a lower
p
1. The unique stable equilibrium is
p
1
=
p
2
=
0 , with agents exclusively receiving the low wages
(
w
1
;
w
2
).
2 When inter-sectorial transfers are symmetric, i.e.
t
1
=
t
2, risk aversion guarantees that condition (4) is veriﬁed.
3For example, in the case of agricultural economies, if
(
1
￿
p
i
) is the probability of a bad crop, the principal in sector
i can affect the risk of
crop failure by investing in fertilizers, machinery, etc. Another interpretation is that
w
i and
w
i are ﬁxed institutional wages for employment in
high- and low-productivity activities, and that
p
i and
(
1
￿
p
i
) are the corresponding employment rates chosen by the principal.
33 Conclusion
This paper has developed a two-sector model to show how inter-agents transfers can affect the distribution of wages.
When the realization of states is observed only by risk-averse agents with uncorrelated income risk, mutual risk-
poolingagreementscanarise. Therentthuscreatedcanbeextractedbyamonopsonistemployerwhochoosesthewage
distribution to maximize the surplus of his employees which he subsequently extracts. One important implication is
a transfers-based inter-sectorial transmission mechanism. This linkage between wage distributions across sectors can
have perverse effects if both sectors are monopsonistic and do not collude: all employment tends to be concentrated
in low-wages allocations.
Finally, an application of our analysis could be rural and urban sectors where it is in the best interest of a unique
rural employer to keep up with the pace of development in the urban sector by increasing rural employment in high
wage activities.
4References
Bardhan, Pranab (1989) The Economic Theory of Agrarian Institutions (New York: Oxford University Press)
Bencivenga, Valerie R., and Bruce D. Smith (1997) ‘Unemployment, migration and growth.’ Journal of Political
Economy 105, 582–608
Kocherlakota, Narayana R. (1996) ‘Implications of efﬁcient risk sharing without commitment.’ Review of Economic
Studies 63, 959–609
Larson, Donald, and Yair Mundlak (1997) ‘On the intersectoral migration of agricultural labor.’ Economic Develop-
ment and Cultural Change 45(2), 295–319
Ligon, Ethan, Jonathan P. Thomas, and Tim Worrall (1997) ‘Informal insurance arrangements in village economies.’
mimeo, Agricultural and Resource Economics, University of California at Berkeley
Pollak, Robert (1985) ‘A transaction approach to families and households.’ Journal of Economic Literature 23, 581–
608
Tomich, Thomas P., Peter Kilby, and Bruce F. Johnston (1995) Transforming Agrarian Economies: Opportunities
Seized, Opportunities Missed (Ithaca: Cornell University Press)
World Bank (1994) Averting the Old Age Crisis: Policies to Protect the Old and Promote Growth (New-York: Oxford
University Press)
5Figure 1:
1
0
1
p
￿
1
p
2
p
1
p
i
￿
P
r
(
w
i
=
w
i
)
;
i
=
1
;
2
p
2
Convex contract set: shaded area is joint distribution
where risk-pooling contracts take place.
p
1
p
2
Poverty trap
0
p
￿
1
p
2
p
￿
2
6

Inter-Sectorial Risk Pooling and Wage Distributions

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Inter-Sectorial Risk Pooling and Wage Distributions

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