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This paper presents a non-cooperative game model of agricultural institutional changein Philippines to explain findings by Hayami and Kikuchi(2000) that harvesting systemhas changed from Hunusan (collecting harvesting) to Gama (contract harvesting) andagain Gama to Hunusan. In the model, the social norm is formulated by assuming theexternality in the utility functions of farmers. It is shown that such institutional changesmay occur not because of the productivity growth but because of the population changes.
JEL Classification Number: O12, O15, Q12, J43 ∗Faculty of Law and Economics, Chiba University. 1-33 Yayoi-cho, Inage-ku Chiba 263-8522 Japan (e-mail: sakakibara@le.chiba-u.ac.jp). I am grateful to Fumihiro Kaneko, Masao Kikuchi and NobuhikoFuwa for their helpful comments.
The purpose of this paper is to present a game theoretic model that explain the agri- cultural institutional changes of harvesting system in Philippines from 1960’s to 1980’s.
In their celebrated book, Hayami and Kikuchi(2000) observed that rice harvesting sys-tem has changed from Hunusan (collecting harvesting) to Gama (contract harvesting)during 60’s and 70’s and again Gama to Hunusan during 80’s in East Laguna villagelocated near Manila, Philippines.
These changes might have emerged as a result of many complex factors. Indeed, during those periods, the village experienced a lot of shocks such as green revolution,land reform, modernization, establishment of irrigation system and migration. Probablyevery such shock might be attributed to these changes in harvesting system. But themajor factor for these changes has been still unknown. Hayami and Kikuchi claim thatthese changes are attributed to the productivity growth and the existence of social normin the village: “the gama contract represents an institutional arrangement designed to reduce disequilibrium between the renumeration rate of labour and its marginal productivity within the framework of work and income-sharing in the community.(p.173)” Accompanied with many empirical findings, their explanation seems to be plausible, butit still remains unclear what the social norm is and how the norm affects the change inharvesting system.
To answer these questions, I construct a multi stage non-cooperative game model in which each farmer’s choice of the harvesting system on his land between Hunusan andGama is given as a Nash equilibrium. In the model Hunusan is the harvesting systemthat permits everyone in the village to participate in harvesting on the farmer’s landand earn a certain share of output. On the other hand, Gama is the one that permitsone laborer to harvest the land exclusively. Also, the social norm is introduced in theway that farmers are sympathetic to the villagers. The sympathy is embodied in formof externality in the utility function of each farmer.
There are two types of agricultural laborers, workers and outsiders, in the model.
Every farmer feels sympathy to the former people as well as the other farmers, butnot to the latter, and his choice of the harvesting system is affected by not only hisown consumption of rice, but also that of workers and other farmers. In his choice ofthe harvesting system, the choice of Gama gives a laborer more amount of rice, but atthe same time it excludes the opportunity of other individuals, including other workersand farmers, to harvest his land by which his utility may decrease through decreases inother villagers’ utilities. In the model, there are three types of subgame perfect Nashequilibrium; i) Hunusan prevails in all lands; ii) Gama prevails in all lands; and iii)Hunusan and Gama coexist.
In literature related to this paper, North(1990) discusses the institutional change and economic growth. Theoretically, from the viewpoint of institutional arrangement in game theoretic framework, the change of institution in the form of the state are discussed byOkada and Sakakibara (1991) and Okada, Sakakibara and Suga (1997) in a dynamicpublic good economy. Also the emergence of property rights on ownership of wealth in aHobbesian anarchy are discussed by Sakakibara and Suga (1997) and Sakakibara(2000).
The main result of this paper is as follows. First, the more number of the outsiders and the stronger the social norm, the more lands are under Gama contract in equilibrium.
And without outsiders or the social norm, no Gama prevails in the resulting equilibrium.
Secondly, the productivity of the economy and the fraction of share of harvesting, though these values affect the consumption level of each individual in the economy, donot matter on the equilibrium number of Gama and Hunusan which depend only on thenumber of individuals of each type. This result is different from the traditional view ofinstitutional change that attributes it mainly to the productivity growth of the economy.
In the model, the productivity growth might affect the resulting equilibrium indirectlythrough the change in the population, but if there exists a competitive labor market,then the effect of growth is completely absorbed in the change of the fraction of shareand consequently has no effect on the equilibrium number of Hunusan and Gama.
By using these features of the model and comparing them to actual observations, it seems that the first change from Hunusan to Gama could be explained as a result of theincrease in the number of outsiders through migration, and that the second change fromGama to Hunusan as a result the social norm to be weakened. This explanation seemsto suggest that the productivity growth affects the institutional change only indirectly,and that the population growth could be a more important factor of it.
The paper is organized as follows. In the following section, I present a game theoretic model and define the equilibrium. In section 3, some propositions and features of theequilibrium are discussed. All proofs of the propositions are given in Appendix. Thenin section 4, the applicability of the model to the historical observations is discussed.
Finally, in section 5, some remarks are given.
Let us consider a three period economy in which there is only one consumption good named rice. There are three types of individuals named farmers, workers and outsiders.
There are n0 farmers, n1 workers and m outsiders. The farmers are descendants of thefounders of the village, and the workers are relatives of the farmers. While the outsidersmigrated recently from outside. Each farmer owns one unit of land that produces yunits of rice in the second period. While the workers and the outsiders do not own anyland. I denote N0, N1 and M as the set of the farmers, the workers and the outsiders,respectively. The utility of each farmer i, ui (i ∈ N0) is given by where ci is the amount of rice consumed by farmer i (i ∈ N0) at the end of the third period.1) The utility of each individual i of workers and outsiders, ui (i ∈ N1 ∪ M), isgiven by where ci is the amount of rice consumed by him at the end of the third period.
Note that there is an externality on each farmer’s utility of the consumption level of other farmers and workers. This externality might be explained as a social norm in thevillage.
There are two harvesting systems, named Gama and Hunusan, in the village. Gama is the contract between a farmer and a worker or outsider that permits the worker oroutsider to harvest the farmer’s land exclusively in the third period. The worker oroutsider receives αy units of rice by this contract, where α (0 < α < 1) is determinedexogenously before the first period. On the other hand, in Hunusan, a farmer opensthe opportunity of harvesting his land to every individual (including himself) in theeconomy.2) I assume that he economy’s harvesting is made made in the following manner.
In the first period, given N0, N1, M , and α, each farmer i (i ∈ N0) chooses the way of harvesting, Gama (abbreviated to G) or Hunusan (abbreviated to H). At the same time,each farmer i who chooses Gama selects an individual ji among workers and outsiders(N1 ∪ M ). If the farmer’s choice is Hunusan (i.e., ai = H) ji is set to be 0. Let ai(ai ∈ {G, H}) be his choice of the way of harvesting. If there is no available workers oroutsiders, then farmer i must choose Hunusan. Let n0(G) be the number of farmers whochoose Gama. Then n0(G) satisfies n0(G) · n1 + m. For simplicity, I assume that thefirst n0(G) farmers chose Gama, i.e., ai = G for i = 1, 2, ., n0(G). Also let n1(G) bethe number of workers selected as a Gama worker. The choice of each farmer i (i ∈ N0)in the first period is given by a pair (ai, ji).
After the selection, i.e., given (ai, ji) (i ∈ N0), each farmer i will ask individual ji whether he will harvest the farmer i’s land in the third period. If the answer is yes, thenthe land of farmer i is harvested exclusively by ji. If the agreement is not reached, thenthe farmer has to change his choice from Gama to Hunusan.
Let ng be the number of farmers who succeeded in making Gama contract after the end of this period. I denote the state of each farmer i (i ∈ N0) at the end of the thirdperiod by a pair (a , j i (i = 1, 2, ., n0(G)) if he succeeded in making Gama contract and a = H if not, and the second term j contracted harvester for the farmer i. If the farmer’s choice is Hunusan (i.e., a = H), as in the first period, ji is set to be 0.
At the beginning of the third period, given {(a , j under Gama and n0 − ng units of land under Hunusan, and each land produces y unitsof rice. If the land of farmer i is under Gama contract, i.e., (a = G), then the output of y units of rice is devided by (1 − α)y for farmer i and αy units of rice for individualj who harvests the land.3) If the land of farmer i is under Hunusan, i.e., a = H, then the land is opened to every individual including himself for harvesting. If the number of individuals who participatein harvesting the land is given by s, then the output of the land y is divided by (1 − α)yfor farmer i and (αy)/s units of rice for each of those who harvest the land.
Let H3 be the set of n0 − ng units of land under Hunusan in the third period, i3 (i3 = 1, 2, ., n0 − ng) be the i3-th element of H3, and h(i3) be the number of individualswho participates in harvesting land i3 ∈ H3. In the economy, every individual j exceptthose who has engaged in harvesting under Gama contract, including every farmer, canparticipate in harvesting on every land under Hunusan. Let Lh(j) ∈ H3 be the set oflands individual j participates in harvesting. Then individual j obtains from each landi3 ∈ Lh(3) the amount of αy/h(i3) units of rice by participating the harvest.
After every land has harvested, the total amount of rice that each individual j obtains which he consumes at the end of the third period, ci, is given as follows. First, ifindividual i works as Gama harvester, then his income is given by αy. Because hecannot participate in Hunusan harvesting, we have Next, if individual i is a worker or outsider who does not make contract as a Gama harvester, his income comes from Hunusan harvesting only. Therefore we have Finally, if individual i is a farmer, irrelevant to whether he chooses Gama or Hunusan, Note that if no workers or outsiders exist, i.e., n1 = m = 0, then, under the above setup, farmers cannot make Gama contracts, and Hunusan prevails in every land in theeconomy.
Under the above setup, we consider a subgame perfect Nash equilibrium.
At the beginning of the third period, there are ng units of land are harvested by ng workers and/or outsiders under Gama. Each individual i who harvests under Gamareceives αy units of rice and does not participate in sharecropping under Hunusan.
Let Wh be the set of n0 + n1 + m − ng individuals who may participate in Hunusan, and let iw be the iw th individual in Wh. At the third period, each individual i∗w ∈ Whdecides the set of lands he participates in harvesting, Lh(i∗w), given the decision of otherindividual iw ∈ Wh, Lh(iw) (iw = i∗w).
If i∗w is a farmer, then his problem is to maximize For each land i3 ∈ H3, let n (i3) be the number of villagers except i∗w who participate in harvesting on i3. Also let h∗(i3) be the number of individuals except i∗w who participatesin harvesting on i3. If he does not participate in harvesting on i3, then his utility gainfrom the harvesting on land i3 is given by α/((h ∗ (i3) + 1)y) + (θn (i3)α)/((h ∗ (i3) + 1)y).
The former is less than the latter if and only if h ∗ (i3) − θn (i3) > 0. Since h ∗ (i3) ≥ n (i3) and θ < 1, he decides to participate in harvesting. This condition is satisfiedfor every land i3 ∈ H3, and he decides to participate in harvesting on all lands underHunusan.
If i∗w is a worker or an outsider, then his problem is to maximize If he does not participate in harvesting on i3, then his utility gain from the harvesting on land i3 is zero. If he does, then it is given by Therefore, obviously, he decides to participate in harvesting on i3. This condition is satisfied for every land i3 ∈ H3. and he decides to participate in harvesting on all landsunder Hunusan. Summing up, every individual except those who harvests under Gamacontract participates in every land under Hunusan, and he obtains from Hunusan thetotal of [(n0 − ng)/(n0 + n1 + m − ng)]αy units of rice.
Let ng1 be the number of workers who participate in Gama. Then the utility of every ui = (1 − α)y + [(n0 − ng)/(n0 + n1 + m − ng)]αy +θ[ng1αy+(n0 − 1)[(1 − α)y + [(n0 − ng)/(n0 + n1 + m − ng)]αy] +(n1 − ng1)[[(n0 − ng)/(n0 + n1 + m − ng)]αy].
For a worker or an outsider i, if he harvests under Gama, then his utility ui is given and if he does not harvest under Gama, then his utility ui is given by ui = [(n0 − ng)/(n0 + n1 + m − ng)]αy.
i=1, every farmer i who has chosen an individual ji, asks individual ji whether he agrees to be a harvester of his land. If individual jiagrees, then Gama is contracted between farmer i and individual ji. If not, then thefarmer i sets his land under Hunusan. Let n0(G) be the number of farmers who offeredGama and let n1(G) the number of workers who has offered Gama in the first period.
Let us consider the decision problem of individual ji who has been selected as a candidate of Gama harvester, given other the decisions of other selected individualsji (ji = j1, j2, .ji−1, ji+1, ., jn0(G)). Other than ji, in the decision, let n0” be Gamacontracts agreed upon and let n1” be the number of Gama harvesters among workers.
If he accepts the offer, then his utility uj is given by αy. If he does not accept the 0 − n0”)/(n0 + n1 + m − n0”)]αy.
It is obvious that individual ji always accepts the offer since the utility under Gama ¿From above, in the second period, given {(a , j Gama agree, and the number of Gama agreed ng is given by n0(G) and the nunber ofworkers under Gama contract, ng1, is given by n1(G).
Now consider the choice problem of farmer i in the first period. Each farmer i either chooses Gama and selects an individual ji as a Gama harvester or Hunusan and doesnot select anyone (ji = 0).
Under the condition that other farmers’ choices (as, js) (s = i, s ∈ N0) be given, I consider farmer i’s problem. Let ng be the number of farmers other than i who selectGama, and let nw be the number of workers whom the farmers other than i select asGama harvesters.
Case 1 If he chooses Hunusan, i.e., ai = H and ji = 0), then total of ng farmers select Gama and nw workers are selected as Gama harvesters. Therefore the utility of farmer i underthe choice of Hunusan, uH , is given by = (1 − α)y + [(n0 − ng)/(n0 + n1 + m − ng)]αy +[(n0 − ng)/(n0 + n1 + m − ng)]αy]+(n1 − nw)[[(n0 − ng)/(n0 + n1 + m − ng)]αy].
If he chooses Gama and select a worker, i.e., ai = G and ji ∈ N1), then total of ng + 1 farmers select Gama and nw + 1 workers are selected as Gama harvesters. Therefore theutility of farmer i under Gama with the choice of a worker, uG1, is given by 0 − ng − 1)/(n0 + n1 + m − ng − 1)]αy +θ[(nw + 1)αy + (n0 − 1)[(1 − α)y + [(n0 − ng − 1)/(n0 + n1 + m − ng − 1)]αy]+(n1 − nw − 1)[[(n0 − ng − 1)/(n0 + n1 + m − ng − 1)]αy].
If he chooses Gama and select an outsider, i.e., ai = G and ji ∈ M ), then total of ng + 1 farmers select Gama and nw workers are selected as Gama harvesters. Thereforethe utility of farmer i under Gama with the choice of a worker, uGm, is given by 0 − ng − 1)/(n0 + n1 + m − ng − 1)]αy +θ[nwαy + (n0 − 1)[(1 − α)y + [(n0 − ng − 1)/(n0 + n1 + m − ng − 1)]αy]+(n1 − nw)[[(n0 − ng − 1)/(n0 + n1 + m − ng − 1)]αy].
[(n0 − ng)/(n0 + n1 + m − ng)] > [(n0 − ng − 1)/(n0 + n1 + m − ng − 1)] Therefore, for farmer i, case 3 is dominated by case 1.
[(n0 − ng − 1)/(n0 + n1 + m − ng − 1)]} ·[1 + θ(−m + ng + nw − 1)].
Therefore farmer i chooses Hunusan in the first period if and only if the above value isnon-negative.
Definition: A subgame perfect Nash Equilibrium is {(a∗, j∗)}n0 i (i ∈ N0), (a∗, j∗) maximizes farmer i’s utility given (a∗ Note that the equilibirum defined above does not depend on the fraction α or the productivity of land y since farmer’s choice depends only on the number of people in thevillage and is independent of such values.
In this section I discuss some features of the equilibrium in relation to the popula- tions, n0, n1 and m, and the strength of sympathy, θ. First we have the following twopropositions.
Proposition 1 If θ = 0, then there exists a unique subgame perfect Nash equilibriumwith (ai, ji) = (H, 0) for all i ∈ N0.
Proposition 2 If n1 = 0, then there exists a unique subgame perfect Nash equilibriumwith (ai, ji) = (H, 0) for all i ∈ N0.
Proposition 3 If m = 0, then there exists a unique subgame perfect Nash equilibriumwith (ai, ji) = (H, 0) for all i ∈ N0.
The above three propositions give necessary conditions for the emergence of Gama in the village. The first condition is the sympathy to other villagers and the second and third one are the existence of poor landless villagers and outsiders, respectively. Iffarmers do not have any sympathy, then he does not choose Gama because, by doing so,he lose the opportunity of harvesting his land. Even though he has a positive sympathyto other villagers, if there is no worker, then the choice of Gama implies to hire anoutsider as the harvester of his land, by which every villager loses the opportunity ofharvesting his land. Also, if there is no outsider, the choice of harvesting system is thematter of income distribution, and it is better for the farmer to choose Hunusan ratherthan Gama. Thus under such conditions, every farmer’s choice is to select Hunusanrather than Gama.
Below I consider the case that these three parameters, n1, m and θ are all positive.
then there exists a subgame perfect Nash equilibrium with n0(G) = n∗g, where n∗g thelargest integer that satisfies the following conditions: i) n1 ≥ n∗g,ii)n0 ≥ n∗gand iii) (m + 3 − 1/θ)/2 ≥ n∗g then there exists a subgame perfect Nash equilibrium with (ai, ji) = (H, 0) for all i ∈ N0.
The above two propositions give conditions for the existence of Gama in the village.
If m is greater than or equal to 1/θ − 1, then Hunusan prevails, and if not, then Gamamay emerge in the village. This critical value 1/θ − 1 solely depends on the sympathy θof farmer to other villagers. And, from condition iii) of Proposition 4, more the numberof outsiders are, the more farmers choose Gama. To see this feature, let us consider thefollowing example.
Let n0 = 100, n1 = 100, θ = 0.1, α = 1/2 and y = 1.
Since 1 − θ(m + 1) = 1 − 0.1 > 0, by Proposition 5, the equilibrium is given by (ai, ji) = { (H, 0) for all i ∈ N0.
Each worker and farmer consume 1/4 units of rice and 3/4 units of rice, respectively.
Since 1 − θ(m + 1) = 1 − 2.1 < 0, n∗g units of land are under Gama. Since (m + 3 − 1/θ)/2 = 13/2, we have n∗g = 6. Therefore the equilibrium is given by Each outsider and worker without Gama contract consumes 47/214 units of rice and each worker with Gama contract consumes 1/2 units of rice. Each farmer consumes77/107units of rice.
case 3: m = 50 Since 1 − θ(m + 1) = 1 − 5.1 < 0, n∗g units of land are under Gama. Since (m + 3 − 1/θ)/2 = 43/2, we have n∗g = 21. Therefore the equilibrium is given by Each outsider and worker without Gama contract consumes 79/398 units of rice and each worker with Gama contract consumes 1/2 units of rice. Each farmer consumes139/199units of rice.
Since 1 − θ(m + 1) = 1 − 21.1 < 0, n∗g units of land are under Gama. Since (m + 3 − 1/θ)/2 = 203/2, we have n∗g = 100. Therefore the equilibrium is given by (ai, ji) = (G, ji) ∈ N1 for all i ∈ N0. Each worker is under Gama contract and con- sumes 1/2 units of rice, while each outsider consumes no rice. Each farmer consumes1/2units of rice.
¿From the above example, we can see that the increase in the number of outsider m causes the increase in the number of Gama contract, and for sufficiently large m, alllands are under Gama contract and all outsiders are excluded from harvesting.
Another parameter other than m that affects the existence of Gama is the sympathy θ. To see this feature, let us consider the following example.
Let n0 = 100, n1 = 100, m = 20, α = 1/2 and y = 1.
Since 1 − θ(m + 1) = 1 − 0.21 > 0, by Proposition 5, the equilibrium is given by Each worker and outsider consume 5/22 units of rice and each farmer consumes 8/11 Since 1 − θ(m + 1) = 1 − 2.1 < 0, n∗g units of land are under Gama. Since (m + 3 − 1/θ)/2 = 13/2, we have n∗g = 6. Therefore the equilibrium is given by Each outsider and worker without Gama contract consumes 47/214 units of rice and each worker with Gama contract consumes 1/2 units of rice. Each farmer consumes77/107units of rice.
case 3: θ = 0.5 Since 1 − θ(m + 1) = 1 − 10.5 < 0, n∗g units of land are under Gama. Since (m + 3 − 1/θ)/2 = 21/2, we have n∗g = 10. Therefore the equilibrium is given by Each outsider and worker without Gama contract consumes 9/44 units of rice and each worker with Gama contract consumes 1/2 units of rice. Each farmer consumes31/44units of rice.
¿From the above example, we can see that the increase in the sympathy θ causes the increase in the number of Gama contract. However, differently from the case of m inExample 1, the number of Gama is bounded above by m + 3. Therefore, if the number ofoutsider m is sufficiently smaller than that of farmer n0, some lands are under Hunusanand outisiders are to earn some amount of rice by harvesting.
In this section I consider the applicability of the model to the actual institutional changes in East Laguna village described by Hayami and Kikuchi(2000). In the model,there are three types of individuals, farmers, workers and outsiders. These types seem tocorrespond to large farmers, all farmers and agricultural laborers, respectively in theirbook. According to their book, the first change occurred as that from Hunusan to Gamain the 60’s and 70’s and the second one as that from Gama to new Hunusan in the 80’s.
In 1950, in the village, Hunusan prevailed on all lands. But during 60’s it had beenchanging to Gama, and in 1976, almost all lands are under Gama. Around 1976, newHunusan appeared and it has been dominating over Gama during 80’s.
First, as for the population, during the first change, net migration to the village of large farmer, small farmer and agricultural laborer are given by -7, 22 and 11, respectivelyin 60’s and -41, -15 and 85, respectively in 70’s. During the second change, the netmigration of them are given by 7, -5 and 7, respectively (p.65, Table 3.10).
As for the first change, it seems to be explained by the model as a result of the increase in the number of agricultural laborers(outsiders) like Example 1. On the other hand, in the second change, the migration was small, it does not seem to affect the change.
However, as Hayami and Kikuchi describes; The hunusan contract that became widespread in the 1098’s and 1990’s was different from the traditional hunusan that prevailed before 1970. In the traditional hunusan that everyone could participate in harvesting and receive an output share, but in the new hunusan only the labourers who received specific invitation from employing farmers were allowed to participate.(p.179) In the above, among the workers and outsiders, those who without “specific invitation” are considered not to be in the set of workers n1 and outsiders m. As it is shown inExample 1, since the decrease in m causes the decrease in Gama, the model mightexplain the second change.
The second change could be explained as the effect of the change in θ. Hayami and Kikuchi explained the second change of Gam to Hunusan as “the shift from thecommunity-type to the market-type of contract”. If such a shift took place in thevillage, it seems to imply the decrease of the sympathy, θ. As it is shown in Example 2,such a decrease would cause the shift from Gama contracts to Hunusan.
In what follows, I consider the effect of green revolution on the institutional changes.
In the model, the productivity y and the share of harvesting α do not affect the resultingway of harvesting, i.e., the choice of Gama or Hunusan, in equilibrium. In this sense,productivity shocks, like green revolution, does not matter. However, the productivityshock might affect the parameters of economy by which the resulting equilibrium mightbe affected.
To consider this, first, suppose there exists a competitive labor market and the real wage ω in terms of rice is applied to harvesting. Then the share of worker αy must beequal to the real wage ω and the increase in y will be absorbed completely through thedecrease of α.
However, such a decrease in α will exaggerate the inequality of income distribution, because all the increase in output will go to the farmer and the harvester’s income willstay the same as before. Such an exaggeration may cause the increase in the sympathyparameter θ, by which more farmers choose Gama in equilibrium. To see this feature,let us consider the following example.
Let n0 = 100, n1 = 100, m = 20, θ = 0.01, α = 1/2 and y = 1.
Since 1 − θ(m + 1) = 1 − 0.21 > 0, by Proposition 5, the equilibrium is given by Each outsider and worker consumes 5/22 units of rice and each farmer consumes8/11 Let n0 = 100, n1 = 100, m = 20, θ = 0.1, α = 1/4 and y = 2.
Since 1 − θ(m + 1) = 1 − 2.1 < 0, n∗g units of land are under Gama. Since (m + 3 − 1/θ)/2 = 13/2, we have n∗g = 6. Therefore the equilibrium is given by Each outsider and worker without Gama contract consumes 47/214 units of rice and each worker with Gama contract consumes 1/2 units of rice. Each farmer consumes184/107 units of rice.
In the above example, by comparing case 1 to case 2, we can see that the productivity shock together with the decrease in α exaggerates the inequality in income betweenfarmers and workers. The farmer’s income is 38/5 times larger than the worker’s onein case 2 while the ratio is 16/5 in case 1. While in case 3, as the sympathy parameterθ increases from 0.01 in case 2 to 0.1 in case 3, the average income of the worker is5154/21400, so that the ratio is 36800/5154 which is smaller than that in case 2.
For the another case, if there is no competitive labor market in the economy, then it might occur that α is incompletely adjusted and the share of harvesting αy becomeshigher than the market wage rate ω. In this case, it seems that the higher share attractspeople outside the economy and cause the increase in the number of outsider m throughmigration. As we have seen in Example 1, the increase in m causes the increase in thenumber of Gama. The increase of Gama decreases the working opportunity of outsidersby which the migration will decrease.
According to the above two cases, the productivity shock seems to increase indirectly the number of Gama. In this sense, the productivity shocks seem to matter on theeconomy.
This paper has presented a game model that explains the institutional changes from Hunusan to Gama and from Gama to Hunusan from the viewpoint of the populationstructure and the social norm in the village. Apparently there are many other factorsthat affect actual historical changes observed on East Laguna village in Philippines.
However, it seems important that such changes could emerge without the productivitygrowth which has been a dominant view of the historical change.
In the model, I assumed identical farmers who have the same amount of land and the same strength of sympathy to other villagers. However, as Hayami and Kikuchidescribed in their book, the structure of land ownership is more complicated than thatassumed in the paper. Also there exists other specification of the social norm than that in this paper by which the result could be different from that in this paper. Suchextensions should be done in future research.
1. For the discussion of the social norm from the viewpoint of Economics, see, for 2. For simplicity, I assume that the farmer can harvest his own land under Hunusan.
This assumption can be modified in the way that he cannot harvest his own land,which seems to be more plausible. However, such a modification affects only theallocation, and does not affect the equilibrium choice of harvesting system.
3. Actually Gama is a contract that incudes free weeding on the land. However, if the labor market is competitive, such free weeding must be compensated by the farmerin some manner. Therefore, for simplicity, the free weeding is not considered in thispaper.
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Sen, A. (1987) On Ethics and Economics, Oxford: Blackwell Publishers Ltd.
I consider farmer i’s problem. Let (a∗s, s∗) with s = 1, 2, ., n let ng be the number of farmers other than i who select Gama and let nw be the numberof workers whom the farmers other than i select as Gama harvesters.
= {[(n0 − ng)/(n0 + n1 + m − ng)] − [(n0 − ng − 1)/(n0 + n1 + m − ng − 1)]}· = {[(n0 − ng)/(n0 + n1 + m − ng)] − [(n0 − ng − 1)/(n0 + n1 + m − ng − 1)]} for any number of ng. Therefore farmer i chooses Hunusan. This completes the proof.
Apparent from case 3 of Step 3 in Solution.
I consider farmer i’s problem. Let (a∗s, s∗) with s = 1, 2, ., n let ng be the number of farmers other than i who select Gama and let nw be the numberof workers whom the farmers other than i select as Gama harvesters.
= {[(n0 − ng)/(n0 + n1 + m − ng)] − [(n0 − ng − 1)/(n0 + n1 + m − ng − 1)]}· = {[(n0 − ng)/(n0 + n1 − ng)] − [(n0 − ng − 1)/(n0 + n1 − ng − 1)]}· which is always positive. Therefore i chooses Hunusan. This completes the proof.
Then there exists some integer n∗ ≥ 1 satisfying [1 + θ(−m + 2n∗ − 3)] < 0.
Let n∗g be the largest number n∗ that satisfyi) n1 ≥ n∗,ii)n0 ≥ n∗and iii) (m + 3 − 1/θ)/2 ≥ n∗.
Then we have {[(n0 − n∗g − 1)/(n0 + n1 + m − n∗g − 1)] − [(n0 − n∗g − 2)/(n0 + n1 + m − n∗g − 2)]}· g is an equilibrium. This completes the proof.

Source: http://web.econ.keio.ac.jp/staff/takakofg/coeconfpapers/Sakakibara.pdf