Fabio Zagonari

English

Research Papers in Economics

INEFFICIENT SOCIAL CONVENTIONS
IN BACKWARD ECONOMIES
Fabio Zagonari
Department of Economics
University of Bologna (Italy)
March, 1995
Classification JEL number: C78, Q15
Abstract
When adopted by each member in a society contracts become conventions: they are focal points
that solve coordination problems. This paper develops an evolutionary model with two distinct finite
populationsofplayers(landlords andlandless)in whichplayersfromdifferentpopulationsarerepeatedly
matched within a period to play a stage game. Choices over the three strategies available (fixed rental,
share-croppingandwagecontracts)areaffectedbyboundedrationality.Inparticular,playersassumetheir
opponents to carry on playing the action they adopted the previous period (inertia hypothesis) so that
mimic the most successful strategy played by their own population mates turns out to be the best strategy
(miopia hypothesis). Experimentation allows players to innovate towards the preferred strategy every
period. The calculation of the eigenvector related to the unit eigenvalue of the Markov matrix stresses
that by better solving the collective action problems and decreasing their attachment to habits, landless
can change the status quo towards the desired contract. The analysis of the relative dimensions of the
basins of attractions highlights that the long-run equilibrium is likely to be the contract preferred by
landlords, unless economic incentives are tuned up in order to drive the society to adopt an alternative
one.There is a large literature on agrarian contracts in village economies that, by assuming the full
rationalityofeconomicagentsand,consequently,theefficiencyofinstitutionstheygivebirthto,highlights
the conditions which could favour one contract to the detriment of others (Otsuka, Chuma and Hayami
1992): sometimes in risk attitude of agents; sometimes in prevailing production technology coupled with
endowment distributions across classes of factor owners; sometimes in transaction costs, in general, and
in supervision costs, in particular.
Whenadoptedbyeachmemberinasociety,however,contractsbecomeconventions:theyarefocal
points(Sudgen1986)that solvecoordination problems.The purposeofthispaperisto highlightthat even
ifthere aresomeagentsnot affectedbyboundedrationalitythatpin-pointPareto-superiorcontracts,when
theyall belong to the weaker class so that theycan not afford to innovate because of their low reservation
income, the society can get stuck in inefficient kinds of contracts. Even if the weaker and more rational
classcandrivethesocietytoadoptmoreefficientcontractsintheshort-runbydevelopingarelativedeeper
innovation attitudes and collective action solving capacities, the conditions which make an agrarian
contract prevail in the long-run turn out to lie in the preferences of the stronger class.
I develop an evolutionary model with two distinct finite populations of players (landlords and
landless) in which players from different populations are repeatedly matched within a period to play a
stage game.
Choices over the three strategies available (fixed rental, share-cropping and wage contracts) are
affected by bounded rationality. In particular, players assume their opponents to carry on playing the
action they adopted the previous period (inertia hypothesis) so that mimic the most successful strategy
played by their own population mates turns out to be the best strategy (miopia hypothesis): I will depict
that by assuming a best reply dynamic process. Experimentation allows players to innovate towards the
preferred strategy every period: a system of three non-linear stochastic difference equations will be
analyzed.
The calculation of the eigenvector related to the unit eigenvalue of the Markov matrix stresses that
by better solving the collective action problems and decreasing their attachment to habits, landless can
change the status quo towards the desired contract in the short-run.
The analysis of the relative dimensions of the basins of attractions highlights that the long-run
equilibrium is likely to be the contract preferred by landlords, unless economic incentives are tuned up
in order to drive the society to adopt an alternative one.
The structure of the paper is as follows.
In section 1, the theoretical framework is introduced.
The short-run and long-run behaviour of the system are provided in section 2 and 3, respectively.
The conclusion appears in section 4.
11.THE THEORETICAL FRAMEWORK
I consider an agrarian context where landlords and landless want to reach an agreement on land
utilisation.Itiscommonknowledgethatthiscanbedonebysigningoneofthethreepreconceivedagrarian
contracts: fixed rental (F), share-cropping (S) and wage contract (W). Moreover, both classes know that
not reaching an agreement forces them to resort to the daily rated labour market, where landlords might
not find the required number of workers at peak seasons and landless might not manage to supply the
desired amount of work at slack seasons.
The existenceof uncertainty and adjustment costs makes theinertia aplausible hypothesis: players
suppose that their opponents will not react instantaneously to their environment.
Theanalysiscarried outatvillagelevelsuggeststhat agentsareinapositionto learnwhichstrategy
is better by observing which has worked well for other people from their own population: the myopia
hypothesis is made.
Players,however, arein apositionto tryto changethe statusquo by choosing anewstrategywhich
needs not to be the best response in the previous period. Experimentation, in particular, can be at random
or aimed at: I will depict the former by assuming a uniform innovation rate (l for landlords and t for
landless); I will introduce the factors h (for landlords) and k (for landless) to catch the fraction of times
agents move their experimentation towards the less preferred contract.
It is worthy noticing here that myopia assumption allows me to depict circumstances where
experimenters immediately switch back to their previous strategy if that generated a higher payoff.
Hence, two kinds of agents are around: the conservative ones, who always play the action that
turned out to be the best in the previous period; the progressist ones, who accept short-run losses and
challenge the status quo by introducing a new contract.
Players’choices over contracts are affected by adaptive preferences or bounded rationality. For
example,onecanthink ofagents preferring acontractbecauseoftheignoranceofincomepatternsarising
from alternatives or liking it better because of its rooted tradition in the economy.
Analysis could be carried out whenever classes’preferences disagree. For the sake of simplicity,
however, it is assumed that in spite economic conditions make share-cropping the best solution to risk
sharing and transaction cost problems, only do landless realize that: landlords best prefer the fixed rental
contract. If the matrix of classes’payoff defined in terms of utility is given by:
2landless
  
F,f R,r R,r
LANDLORDS R,r S,s R,r
R,r R,r W,w
the classes’preferences are caught by assuming and . R (with ) and r
(with ) are the standard of living that landlords and landless can ensure themselves, respectively,
when they resort to the daily rated labour market.
Actions are taken in discrete time in such a way that at the beginning of each period t each player
chooses his pure strategy for the period: actions are assumed to be fix within the period. Moreover, each
player assumes to be matched with a player from the opponent class only once.
Let , and the fraction of landless adopting strategy F, S and W, respectively.
Let , and the fraction of landlords adopting strategy F, S and W, respectively.
Therefore, is the average payoff of landless playing strategy c (c= , , ):
Analogously for landlords.
The inertia and myopia hypothesis made above drive me to assume the best reply dynamic:
cF cS cW
cF
cS
cW
s > w ³ fF > S ³ WR < W
r < w
t
F t
S t
W
l
F l
S l
W
pt
c(.) cF cS cW
pt
F(l
F)=fl
F +r(1-l
F)
pt
S(l
S)=sl
S +r(1-l
S)
pt
W(l
W)=wl
W +r(1-l
W)
3if
if
if
Analogously for landlords.
The experimentation hypothesis leads to specify three non-linear stochastic difference equations:
where depict jumps from contact to contract and have binomial distributions.
When :
When :
t
c = 1 pt
c(.) > pt
c(.) for every c ¹ c
bt(l
F,l
S,l
W)= t
c pt
c(.) = pt
c(.) for every c ¹ c
t
c = 0 pt
c(.) < pt
c(.) for some c ¹ c
bl(t
F,t
S,t
W)
tt +1
F = b(lt
F,lt
S,lt
W)-vFS -vFW +vSF +vWF
tt +1
S = b(lt
F,lt
S,lt
W)-vSF -vSW +vFS +vWS
tt +1
W = b(lt
F,lt
S,lt
W)-vWF -vWS +vFW +vSW
vc,c c c
tt
S = 1
vSW = vSF~Bin(b(lt
F,lt
S,lt
W),t/k)
vFW = vWF~Bin(1-b(lt
F,lt
S,lt
W),t/k)
vFS = vWS~Bin(1-b(lt
F,lt
S,lt
W),t)
tt
F = 1
vFS~Bin(b(lt
F,lt
S,lt
W),t)
4When :
Analogously for landlords.
Therefore, an aimed innovation is depicted by an increase of k ( ) for landless and h ( )
for landlords.
This dynamic system defines a Markov chain whose transition probabilities are given by:
and the Markov matrix by:
P embodies the assumption that landless (landlords) innovate towards S (F) a greater number of
times k (h) than they do towards other contracts.
vFW~Bin(b(lt
F,lt
S,lt
W),t/k)
vWS~Bin(1-b(lt
F,lt
S,lt
W),t)
vWF = vSW = vSF~Bin(1-b(lt
F,lt
S,lt
W),t/k)
tt
W = 1
vWS~Bin(b(lt
F,lt
S,lt
W),t)
vWF~Bin(b(lt
F,lt
S,lt
W),t/k)
vFS~Bin(1-b(lt
F,lt
S,lt
W),t)
vSF = vFW = vSW~Bin(1-b(lt
F,lt
S,lt
W),t/k)
k ³ 1 h ³ 1
p(c,c),(˜ c, ˆ c) = Prob(lt +1
c Ù tt +1
c | lt
˜ c Ù tt
ˆ c)
P =[p(c,c),(˜ c, ˆ c)]
5Let be the 3-dimensions simplex where
is the proportion of landless playing strategy c.
Let be the 3-dimensions simplex where
is the proportion of landlords playing strategy c.
Therefore, spec-
ifies the pairs of proportions playing a particular pair of strategy .
2.SHORT-RUN BEHAVIOUR
By construction, all elements in matrix P are strictly positive: the Markov chain has a unique
stationary distribution satisfying:
where can be interpreted as the proportion of time that the society spends on each state (Karlin
and Taylor 1975).
If one normalizes the probability distribution over the possible states at time t=0 such that:
where is the probability not to reach an agreement, the probability to be in state SS at time t=1 is
given by:
Afirstinspectionshowsthat landlesscanaffecttheprobabilitytobeinthepreferredstateinalinear
fashion by increasing either k or t. In particular, for , if:
D3
t º{ t
c Î R
3 | t
c ³ 0 for c = F,S,W & åct
c = 1}
t
c
D3
l º{ l
c Î R
3 | l
c ³ 0 for c = F,S,W & åcl
c = 1}
l
c
D3
2 º{ ( l
c,t
c)ÎC
3 | l
c ³ 0 t
c ³ 0 for c = F,S,W & åcl
c = 1 åct
c = 1}
(c,c)
m=( m FF,m FS,m FW,m SF,m SS,m SW,m WF,m WS,m WW)ÎD 3
2
mP =m
m
m(0)=( 1-m SS(0)-m WW(0)-6m,m,m,m,mSS(0),m,m,m,mWW(0))
m
mSS(1)=
lt
h
+m
æ
ç
è
2
l
h
+2t -6
lt
h
-2lt -4
lt
hk
ö
÷
ø
+m SS(0)
æ
ç
è
1-l -
l
h
-2
t
k
-
lt
h
+2
lt
hk
+2
lt
k
ö
÷
ø
m=0 mSS(1)>m SS(0)
6A stagnant society where landlords defend the status quo is better depicted by the first condition.
If this is the case:
Therefore,landlesscanshowalimitedcapacityinsolvingcollectiveactionproblems(smallk)only
when they are willing to break traditional institutions (large t). Moreover, the better the initial condition
(large ), the greater the class cohesion they have to express (large k).
It is worth noticing that:
The first conditions says that landless have to rely to a greater extent on a reduction of their
attachment to habits (large t) than on an increase in their capacity in solving collective action problems
(large k) when landlords are well organized in defending the status quo (large h).
k
* >
2tmSS(0)(l -h +hl)
l[mSS(0)(1+h +t)-t]
when l <
h
1+h
k
* <
2tmSS(0)(l -h +hl)
l[mSS(0)(1+h +t)-t]
when l >
h
1+h
dk
*
dt
< 0 Û l <
h
1+h
dk
*
dm(0)
> 0 Û l <
h
1+h
m(0)
dk
*
dh
< 0 Û l > 1-
m(0)
t(1-m ( 0))
dk
*
dl
> 0 Û h >
t(1-m ( 0))
m(0)
-1
7The second condition claims that when landlords show a high propensity to innovate (large l),
landless can drive the society to adopt the most efficient contract in the short-run by increasing their
capacity in solving collective action problems (large k).
3.LONG-RUN BEHAVIOUR
When P is strictly positive, the following stability condition holds:
Theequilibriumwhichwillprevailinthelongrun,therefore,doesdependontherelativemagnitude
of the basins of attractions only (Kandori, Mailath and Rob 1993). All the three basins I deal with have
two edges. In order to identify the largest one, I will therefore consider the shortest edge for each basin
and compare them each other: the largest edge will belong to the largest basin.
Share-cropping (SS) will prevail whenever the following set of conditions holds (See Figure 1).
For any q ÎD 3
2 qP
t =m as t ®¥
0 £ R < W
R <
FW -S
2
F +W -2S
r <
sw - f
2
s +w -2f
R <
S(f -r)-F(w -r)
w - f
8Under the preference structure assumed above, the only condition that may not be satisfied is the
lastone.Provided ,thisrequiresthatthelandlords’reservationincomeislowerthansomepositive
amount: it is always possible, therefore, to tune the economic incentives R and r in order to make
share-cropping prevail in the long-run.
4.CONCLUSIONS
When adopted by each member in a society, agrarian contracts may become conventions: they are
focal points that solve coordination problems. This paper highlights that even if each class can change
the relative permanence of the different contracts in the short-run by reducing its attachment to habits
and/orincreasingitsinternalcoordination,forgivenbehaviourbytheopponents,thelong-runequilibrium
islikelytobethecontractpreferredby thestrongerclass,unlesseconomicincentivesaretunedupinorder
to drive the society to adopt an alternative one.
f > w > r
9REFERENCES
KANDORI, M., G.J.MAILATH AND R.ROB, 1993, Learning, Mutation, And Long Run Equilibria In
Games, Econometrica, 61, 29-56
KARLIN, S. AND H.M.TAYLOR, 1975, A First Course In Stochastic Processes, Second Edition, San
Diego, Academic Press
OTSUKA,K.,H.CHUMAANDY.HAYAMI,1992,LandAndLabourContractsInAgrarianEconomies:
Theories And Facts, Journal of Economic Literature 30, 1965-2018
SUDGEN, R., 1986, The Economics of Rights, Cooperation and Welfare, Oxford, Basil Blackwell
10Figure 1
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Inefficient Social Conventions in Backward Economies

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