Informal insurance in renewable resource harvesting
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(2) 220. Rabindra Nath Chakraborty. Sharing rules differ from sharecropping arrangements in that all sharing partners are independent producers. As a consequence, it is reasonable to assume that all sharing partners are risk-averse, especially in settings where the producers are small peasants. When sharing rules are symmetric (i.e. when each producer gives an identical share of his harvest to all others) and producers are identical, risks are shared equally. In a sharecropping context, by contrast, the means of production are owned by one party (the landlord) whereas the tenant contributes labour. As a consequence, it is reasonable to assume that the landlord is much less risk-averse than the tenant. When complete contracts can be written, it is then efficient that risks are shared unequally. For example, a risk-neutral landlord assumes all the risk and pays the tenant a fixed wage (Stiglitz, 1974). However, it is (second-best) efficient that the tenant assumes part of the risk when the tenant’s labour input cannot be observed by the landlord. In the model of sharing rules presented in this paper, a somewhat analogous effect emerges. Although risks are shared equally among producers, the unobservability of labour inputs generates a moral hazard effect that prevents the producers from sharing their risks fully when the resource stock is small. Sharing rules also differ from crop insurance schemes because the payment that a sharing partner receives is not contingent on the realisation of their income whereas the payment of the insurance premium is. In this sense, sharing rules have characteristics of both an insurance contract and a redistributive tax. The interpretation of sharing rules as an informal insurance contract in this paper is (to the best of my knowledge) new. Sharing rules have existed in many different societies and regions in the world. For example, game is shared among the members of a successful hunting party in most hunter–gatherer societies (Woodburn, 1982; Ka¨gi, 2001). Wiessner (1977) emphasised that the system of reciprocal gift giving among !Kung San hunter–gatherers in southern Africa can be interpreted as a risk-sharing institution. In the coastal fisheries of the Pacific region, a rule is widespread whereby each fisherman must share his catch with other residents on his island (Mauss, 1923/1954; Alkire, 1977; Ruddle, 1996; Bender, 2001). Obviously, such rules have increased the chance of survival in traditional societies because they enable individuals to smooth consumption over time. Moreover, sharing rules may also have increased the chance of survival because they support resource conservation. Evidence from the Kingdom of Tonga suggests that sharing rules indeed have a positive impact on resource conservation. Bender et al. (2002) compare coastal subsistence fisheries around two islands in the Ha’apai region of Tonga that differ in their resource stock levels. Whereas fish stocks are lower in the fishing areas of the island of U’iha, stocks are larger around the other island, Lofanga. At the same time, the authors find sharing rules to be stronger (i.e. average insurance premia to be higher) among the population of Lofanga than on U’iha. On both islands, producers share less than all of their incomes. The authors put forward the hypothesis that informal insurance causes moral hazard in the sense that fishermen expend less effort on fishing, which helps to conserve fish stocks..
(3) Informal insurance in renewable resource harvesting. 221. In this paper, an economic model is developed that is consistent with the Bender et al. (2002) hypothesis. The model is based on a general equilibrium framework that was originally developed by Brander and Taylor (1998). It differs from the Brander–Taylor model by assuming a constant population and a stochastic harvesting technology. We consider an agrarian economy where no human-made capital is accumulated. Economic activity takes the form of renewable resource harvesting and an alternative activity called ‘manufacturing’. Manufactures are produced with constant returns to scale to labour whereas harvesting features constant returns to scale in labour input and in the resource stock. The resource exhibits logistic growth. An individual’s harvest attains a fixed positive level with probability and zero with probability 1 . Consumers maximise instantaneous utility to divide their consumption expenditure among resource units and manufactures. Producers, who are assumed to be risk-averse, maximise the certainty equivalent of their incomes from resource harvesting and manufacturing under perfect competition. A general equilibrium approach is adopted for the following reasons. First, it permits an analysis of the incentives for sharing under both production and price risks, as producers’ incomes are correlated through the price formation process even when individual yields are not. Second, it takes into account that consumers can smooth their utility through the market purchase of goods, which is feasible in most agrarian societies. The general equilibrium approach therefore permits the analysis of the relationship between insuring risks through sharing rules and through the goods market. Third, the general equilibrium approach presented here is an adequate analytical description of the economic setting encountered by Bender et al. (2002) in the agrarian economy of Lofanga Island, where fisheries and horticulture are the most important economic activities. The results of the model are as follows. An increase in the insurance premium causes moral hazard in the sense that individuals expend less effort on resource harvesting. However, the resulting inefficiency is reduced by an increase in the expected level of the resource stock (and, hence, expected harvest) at the stochastic long-run equilibrium. A statically (second-best) efficient level of sharing exists, which is positive but smaller than the firstbest level if the resource stock per capita is small. In a decentralised setting, an exogenous increase in the labour force lowers individual harvesting effort if the resource stock per capita is small. This paper is organised as follows. Section 2 presents the model. Section 3 analyses the properties of the model at a given resource stock level, and Section 4 briefly characterises the impact of the dynamics of the resource stock. Section 5 discusses two generalisations with regard to the probability structure of the model. Section 6 concludes.. 2. The model We consider an economy where two goods are produced: a manufactured good and the harvest of a renewable resource. The manufactured good.
(4) 222. Rabindra Nath Chakraborty. (denoted M) can be interpreted as an index of agricultural or artisanal goods and reproductive activities (child-rearing, family duties, and so on). It is produced with a fixed coefficient technology that uses only labour. One unit of labour is assumed to produce one unit of M. As both labour and goods markets are perfectly competitive, the wage rate w equals one unit of manufacturing output if manufactures are produced. The harvest of the renewable resource in this model can be interpreted as any risky economic activity that depends on the extraction of a renewable resource. An obvious example is fisheries. Furthermore, resource harvesting can take the form of agriculture insofar as the soil is exploited as a renewable resource and the productivity of its exploitation is subject to uncertainty. Resource users are assumed to employ a stochastic harvesting technology. It is assumed that the realisation of the ith producer’s harvest rate hPit at time t positively depends on the current resource stock St and the amount of labour litH allocated to resource harvesting: hPit ¼ cit St litH :. ð1Þ. cit 0 is a random variable that determines the productivity of resource harvesting at a given resource stock and labour input. It is assumed that the probability distribution of c is identical for all producers, statistically independent across producers, and constant over time. It is further assumed that c equals z > 0 with probability and zero with probability (1 ). One may think of a number of fishing vessels that cover a large fishing area. Each vessel can catch a defined quantity of fish, zSt litH , at a given level of labour input and the resource stock, and each unit of fish comes close to any fishing vessel with equal probability. Labour input can be interpreted both in physical (time) and efficiency units. If counted in efficiency units, differences in labour inputs can account for differences in fishing technologies, e.g. handlining versus spearfishing. The harvest function just specified may be considered as restrictive on two grounds. First, increased labour input in harvesting may involve searching activities that raise the probability of a successful catch. In this context, it is possible to consider as a function of labour input, ¼ ðlitH Þ, d=dlitH > 0. Second, the catches of individual fishing vessels are correlated in many realworld situations. However, Section 5 shows that the main results of the model do not depend on the assumptions of stochastic independence and a constant value of . Under the simplifying assumptions made here, the expected individual resource harvest EðhPit Þ and the variance of individual harvest at a given resource stock level St and labour input litH can be computed from (1) as EðhPit Þ ¼ zSt litH. ð2Þ. VarðhPit Þ ¼ z2 ð1 ÞðSt litH Þ2 :. ð3Þ. The labour force is constant over time and consists of L > 1 identical individuals, each of whom is endowed with one unit of labour. Each individual is.
(5) Informal insurance in renewable resource harvesting. 223. both producer and consumer. It is further assumed that the number of resource users is small, so that the law of large numbers is not applicable. The aggregate expected harvest EðHtP Þ supplied by producers is then X L hPit ¼ L EðhPit Þ ¼ zSLl H ð4Þ EðHtP Þ ¼ E i¼1 H. with Ll being the aggregate amount of labour allocated to resource harvesting. It is further assumed that an informal insurance scheme exists, which works as follows. Resource users with positive harvests pay an insurance premium that is a fraction of their harvest in value terms. Aggregate premium payments are redistributed equally among all resource users. It is assumed that individual resource harvests are observable whereas individual labour inputs are not. As a consequence, the sharing rule is enforced by the resource harvesters monitoring each other during their daily activities whereas individual labour inputs are not contractible. The scheme just described combines characteristics of both a resource tax and insurance contracts. Obviously, it achieves a pooling of risks, which is a feature of insurance contracts. However, the insurance payments are not state contingent, as the aggregate premium is distributed both among successful and unsuccessful resource users, which is a characteristic of a redistributive tax. The resource is harvested under open access. As perfect competition prevails in the goods market, the resource price adjusts instantaneously to equate aggregate supply and demand of the resource harvest:3 L X i¼1. hPit ¼. L X. hD it ð pt Þ. ð5Þ. i¼1. with hD it ð pt Þ being the realisation of individual i’s demand for the resource at time t, which is a function of the resource price, pt . As harvest is stochastic, the resource price is stochastic, too. Although individual harvest rates have been assumed to be statistically independent, it needs to be emphasised that income risks are correlated among resource users because the resource price and sharing incomes depend on aggregate resource harvest. In other words, resource users face both systemic and idiosyncratic income risks. Individual consumers choose levels of individual demand for the resource, hD it , and for the manufactured good, mD , to maximise instantaneous utility u : t it D 1 ut ¼ ðhD it Þ ðmit Þ. with 0 < < 1. ð6Þ. subject to the budget constraint D hD it þ mit =pt ¼ Iit. ð7Þ. with Iit being the realisation of the income of the ith consumer at time t, measured in units of the resource. Obviously, the existence of a goods market 3. As the number of resource users has been assumed to be small, this outcome is also consistent with Bertrand competition among the resource harvesters..
(6) 224. Rabindra Nath Chakraborty. enables consumers to smooth their consumption of the resource when an unsuccessful catch causes their real incomes to decline. Producers allocate labour between resource harvesting and the production of manufactures. The following analysis assumes that producers are riskaverse in the sense of expected utility theory (Pratt, 1964; Arrow, 1970). Their utility function is UðIÞ; it is assumed that U 0 ðIÞ > 0 and U 00 ðIÞ < 0. As they know the volume of their (stochastic) harvest only after their harvesting effort has been expended, it is reasonable to assume that they base their labour allocation decision on the certainty equivalent of their income, given the resource price and the level of the resource stock. An individual producer chooses his labour input to resource harvesting, liH , to maximise the certainty equivalent of his income, I:4 I ¼ EðIÞ VarðIÞ=2 ð8Þ where EðIÞ is the expected value of income, the coefficient of absolute risk aversion, and VarðIÞ the variance of the producer’s income (Pratt, 1964). The coefficient of absolute risk aversion is assumed to be constant over the relevant range. This specification is adopted because it simplifies the analysis and corresponds to a second-order approximation of any von Neumann– Morgenstern utility model. The ith producer’s realised income is L X w H P ð9Þ I ¼ Iit ¼ ð1 lit Þ þ hit 1 þ hPjt : þ pt L L j¼1; j6¼i The first term in (9) is income from manufacturing (measured in resource units), which is not subject to production risk but is subject to fluctuations in the resource price. The second term represents individual harvest minus the insurance premium, , plus the fraction =L of producer i ’s insurance premium that is ‘paid back’ to that producer through the sharing process. The third term is the quantity of resource units the producer receives from the redistribution of the aggregate insurance payments (excluding his own). Equation (9) illustrates that individual producers face uncertainty not only with regard to his own harvest rate but also with regard to the harvest rates of all other producers, which jointly determine the resource price and the insurance payment (i.e. the quantity of resource units that is shared). The dynamics of the renewable resource are described by a logistic growth function in discrete time. The change in the resource stock between time t and t þ t is L X S St þ t St ¼ rSt 1 t t hPit t: ð10Þ K i¼1. 4. The time subscript t is employed to denote realised (as opposed to planned) values of harvesting effort..
(7) Informal insurance in renewable resource harvesting. 225. The first term on the right-hand side of (10) represents the own rate of growth of the resource, i.e. the rate at which the resource regenerates if it is undisturbed by human intervention. The ‘speed’ of regeneration depends on the resource stock St and a parameter r > 0 that is called the ‘intrinsic growth rate’ of the resource. The second term represents the aggregate rate of harvest in time interval t. The change in the resource stock per unit of time, ðSt þ t St Þ=t, becomes negative if St exceeds K, which implies that K is the maximum possible resource stock (‘carrying capacity’).. 3. Short-run equilibrium It is assumed that the resource stock changes slowly whereas prices, labour allocation decisions, and production adjust instantaneously. Hence, the level of the resource stock can be treated as given in the short run. The following section establishes the existence of the resulting short-run equilibrium, and Sections 3.2 and 3.3 consider its comparative statics and efficiency properties. 3.1. The Cournot–Nash solution at a given insurance premium 3.1.1. Price formation. Maximisation of (6) subject to (7) yields the individual demand functions D mD it ¼ ð1 Þpt Iit and hit ¼ Iit . It follows that manufactures are produced at any positive level of income. At equilibrium, all producers choose identical levels of labour input to harvesting, i.e. litH ¼ ltH . With (1) and (9), market clearing condition (5) can be rewritten as L 1 S l H ð1 Þ X ¼ tt c ¼ pt wLð1 ltH Þ i ¼ 1 it. L X. cit. ð11Þ. i¼1. with. St ltH ð1 Þ : wLð1 ltH Þ. The expected value of the price of the manufactured good (measured in resource units), Eð1=pt Þ, is 1 S l H ð1 Þ E zL ¼ zL: ð12Þ ¼ tt pt wLð1 ltH Þ 3.1.2. Labour allocation. It is assumed that the ith individual producer varies liH to maximise Ii while treating the labour inputs ljH of all other producers as constant. The expected value of the ith producer’s income can be computed as a function of liH from (9), (12), and (2): L X EðIÞ ¼ zLwð1 liH Þ þ zSt liH 1 þ EðhPjt Þ: ð13Þ þ L L j ¼ 1; j 6¼ i.
(8) 226. Rabindra Nath Chakraborty. Using (11), the variance of individual income can be expressed as 2 2 H VarðIÞ ¼ z ð1 Þ f þ St li 1 þ L 2 H H þ ðL 1Þ wð1 lj Þ þ St lj : L. ð14Þ. The term f wð1 litH Þ represents the impact of producer i’s labour allocation decision on the variance of the price for manufactures. An increase in litH increases the level and the variance of the price for manufactures. The producers maximise (8) with respect to liH subject to (13) and (14). Each resource user considers the current level of the resource stock, the price for manufactures, and the labour inputs of all other producers as given. The resource user has no incentive to base his resource allocation decision on the expected longrun resource stock level because the resource is under open access. Consequently, the resource user treats EðhPjt Þ; , f, and ljH as constants. The first-order condition is dI H ¼ gðl ; Þ. zwL þ zS z2 ð1 Þ 1 þ i t L dliH ð15Þ H f þ St li 1 þ St 1 þ ¼ 0: L L At the resulting Cournot–Nash equilibrium, all producers choose identical levels of liH ¼ l H . With 1 ð1=LÞ, the first-order condition takes the following form: 1 1 l H ð1 Þ 1 þ 1 ¼ 1 l H zSt ð1 Þ : ð16Þ L ð1 l H Þð1 Þ The right-hand side (RHS) of equation (16) represents the marginal income from the production of manufactures per unit of expected marginal net income from resource harvesting. The RHS rises from zero to infinity as l H rises from zero to one. The left-hand side (LHS) represents the marginal certainty equivalent of income from resource harvesting, normalised by the expected marginal net income from resource harvesting. The LHS equals one for l H ¼ 0 and declines to a finite value as l H approaches one. It follows that a value of l H ¼ l H with l H 2
(9) 0;1½ exists that satisfies (16). Figure 1 illustrates this result. The LHS of (16) is represented by the straight line AA whereas the RHS is represented by the curve OD. Furthermore, equation (16) permits an analysis of the impact of systemic and idiosyncratic risks. The first term in square brackets represents systemic risk, which takes the form of the price risk here. The second term represents (idiosyncratic) individual harvest risk. The third term, , represents the impact of sharing. Sharing reduces the marginal variance by a less than proportionate amount when the number of resource users is finite (as < 1). If the number of resource users increases without bounds, the systemic.
(10) Informal insurance in renewable resource harvesting. 227. Figure 1. Labour allocation at a given resource stock level.. risk declines to zero and sharing reduces marginal variance proportionately. This corresponds to a situation where the law of large numbers is applicable. 3.2. Comparative statics This section analyses the impact of a marginal increase in the insurance premium, , and the number of resource users (labour force), L, on the allocation of labour between resource harvesting and manufacturing. 3.2.1. The impact of the insurance premium. Implicit differentiation of (16) reveals that dl H =d < 0 if 1 þ 2ð1 Þ < 0: 1 þ l H zSt ð1 Þ L The sign of @l H =@ is negative if the decline in marginal variance caused by a rise in (represented by the second term in curly brackets) is smaller than the decline in marginal expected net harvest income (normalised to 1 in curly brackets). This is the case if the maximum harvest per capita, l H zSt , is small. A sufficient condition for @l H =@ < 0 is that the resource stock per capita is small. To see this, it should be noted that the highest possible aggregate resource harvest cannot exceed the resource stock, i.e. LzSt l H St . Consequently, z 1=Ll H . If z attains its maximum value, harvest per capita equals the resource stock per capita, St =L. If the resource stock is small relative to the population, a marginal increase in the insurance premium lowers the amount of labour allocated to resource harvesting. The reason is as follows. First, an increase in the insurance premium lowers a producer’s expected net income from resource harvesting, zl H St ð1 Þ. This decreases the marginal return to harvesting effort, which makes it attractive to reduce l H . Second, an increase in the insurance premium lowers the contribution of the producer’s own harvest to income variance,.
(11) 228. Rabindra Nath Chakraborty. which raises the certainty equivalent of the producer’s income and makes it attractive to increase l H . As the second effect rises with the resource stock per capita, the first effect dominates if the resource stock per capita is small. 3.2.2. The impact of the labour force. Implicit differentiation of (16) reveals that dl H =dL < 0 if 1 1 l H ð1 Þ H 1þ l zSt ð1 Þ 2 < 0: L ð1 l H Þ½1 þ ð=LÞ
(12) 2 L2. ð17Þ. It should be noted that ð1=Þ 1 þ > 0 because < 1. Observing that z 1=Ll H , it can be seen by inspection that the LHS of (17) is negative if the resource stock per capita is small. It follows that @l H =@L is negative if the resource stock per capita is small. This outcome is caused by two forces. First, an increase in the labour force raises the marginal certainty equivalent of income from resource harvesting because it lowers income variance, which is represented by the positive sign of the first term in (17). Second, it reduces the fraction of the insurance premium that is ‘paid back’ to the individual producer, =L, which lowers harvesting incentives (represented by the negative sign of the second term in (17)). The second effect dominates when the resource stock per capita is small. 3.3. Efficient sharing rules The level of sharing (measured by the insurance premium ) was treated as exogenous in the previous sections. This section investigates the incentives producers face when choosing a particular level of sharing. More precisely, it considers the properties of statically efficient sharing rules under two sets of conditions. Section 3.3.1 considers the first-best case where individual labour inputs are assumed to be contractible (contrary to the assumption made above). It will be shown that incentives exist for producers to share their entire harvest in this case ð ¼ 1Þ. If labour inputs are non-contractible, however, it may be efficient for producers to share less than all of their harvests because an increase in the insurance premium lowers harvesting effort. It is then rational for producers to agree on an insurance premium that maximises the certainty equivalents of their incomes under the Cournot–Nash solution. This case is considered in Section 3.3.2; the resulting insurance premium is called second-best efficient. 3.3.1. The first-best case. A first-best solution would entail maximisation of I with respect to and l H ¼ liH , i ¼ 1; . . . ; L while treating EðhPjt Þ in (13) as a variable. As EðhPjt Þ ¼ zSt l H at equilibrium, the derivative @ I=@ reduces to @ I 1 2 H 2 ¼ z ð1 ÞðSt l Þ 1 ð1 Þ 0: ð18Þ @ L.
(13) Informal insurance in renewable resource harvesting. 229. Obviously, I is maximised if is set equal to one. With ¼ 1, the first-order condition for l H can be expressed as 1 l H zSt ð1 Þ. 1 l H ð1 Þ ¼ : L ð1 l H Þ. ð19Þ. The RHS of equation (19) rises from zero to infinity as l H rises from zero to one. The LHS equals one for l H ¼ 0 and declines to a finite value as l H approaches one. It follows that a value of l H ¼ l with l 2
(14) 0;1½ exists that satisfies (19). It is useful to compare the first-best solution with the decentralised solution derived in Section 3.1.2. When is set equal to one, the LHS of (16) and (19) are identical. However, the RHS of (16) exceeds the RHS of (19) by factor L for any value of 0 < l H < 1. This implies that l > l H ð ¼ 1Þ. Geometrically, the OD curve in Figure 1 shifts down, which generates the new curve OF. With full sharing ð ¼ 1Þ, producers in the Cournot–Nash equilibrium allocate less effort to resource harvesting than would be efficient. The same is true if no sharing takes place at all ð ¼ 0Þ. Now the RHS of (16) and (19) are identical whereas the LHS of (16) is smaller than the LHS of (19). The effect just described is caused by the fact that an individual producer takes into account the impact of his labour allocation decision on the expected sharing income that is paid back to him but ignores its impact on the sharing incomes paid to others. In other words, the producer does not consider the impact of a marginal collective change in labour inputs on his own expected sharing income. As individual labour inputs cannot be observed by others, the resulting inefficiency represents a moral hazard effect. 3.3.2. The second-best case. The second-best solution maximises I with respect to and l H subject to the individual labour allocation decision (15) and the constraints l H , 2 ½0;1
(15) while treating EðhPjt Þ in (13) as a variable. Condition (15) can be interpreted as an incentive compatibility constraint. The Lagrangian is L ¼ I þ gðliH ; Þ þ 1 ð1 l H Þ þ 2 ð1 Þ and , 1 , and 2 are the multipliers associated with the constraints. Kuhn–Tucker conditions are @L @ I @g ¼ H þ H 1 0 H @l @l @l @L @ I @g ¼ þ 2 0 @ @ @ @I @g @I @g H þ 2 ¼ 0 þ H 1 l þ @ @ @l H @l. The. ð20Þ ð21Þ ð22Þ. gðliH ; Þ ¼ 0 1 ð1 l H Þ þ 2 ð1 Þ ¼ 0. ð23Þ. H. ð24Þ. 1 ; 2 ; l ; 0 H. ð1 l Þ 0; ð1 Þ 0:.
(16) 230. Rabindra Nath Chakraborty. As I is strictly concave in l H and , a global maximum must exist that satisfies these conditions. The following Proposition states that the second-best efficient level of sharing is positive but smaller than one ð0 < < 1Þ if the resource stock per capita is small. If the resource stock per capita is large, it is efficient for the resource users to share their entire catch ð ¼ 1Þ; however, the resulting allocation differs from the first-best solution in that the effort level is l H < l . The reason is that, at high resource stock levels, a marginal increase in the insurance premium reduces variance more strongly than it lowers effort. Proposition 1. The insurance premium that maximises I subject to (15) is positive and smaller than one ð0 < < 1Þ if the resource stock per capita is small. It is equal to one ð ¼ 1Þ if the resource stock per capita is high. Proof. See Appendix.. 4. Long-run equilibrium The previous section analysed the solution of the model for a given point in time. However, the resource stock changes over time according to equation (10). As the resource harvest is stochastic, the dynamics of the resource stock is stochastic, too. If the continuum of resource stock levels is divided into n finite states, the dynamics of the resource stock can be described as a Markov chain that converges to a limit distribution of states under certain conditions. The limit distribution of states can be used to compute an expected resource stock level. It can be demonstrated that the level of the expected resource stock rises as the insurance premium increases. That is, the static moral hazard inefficiency is reduced by a dynamic gain, which results from the (unintended) partial internalisation of the common pool externality that is brought about by the decline of harvesting labour inputs. Chakraborty (2001a) contains a formal proof.. 5. Extensions This section considers two extensions to the model presented above. The first treats the probability of a successful catch, , as being an increasing function of labour input. The second treats individual catches as being (positively or negatively) correlated. The following analysis demonstrates that the major static results of the model, i.e. the existence of a moral hazard effect and the inverse relationship between the insurance premium, , and equilibrium labour input, l H , for small levels of the resource stock per capita continue to hold under these more general conditions unless the covariance among yields is very high. 5.1. Effort affects probability Let us consider first the Cournot–Nash solution of Section 3.1.2. Producers maximise the certainty equivalent of their incomes with expected income and income variance given by (13) and (14), respectively. However, is.
(17) Informal insurance in renewable resource harvesting. 231. now a function of labour input, ¼ ðlitH Þ, 0 d=dlitH > 0. To keep the analysis simple, it is assumed that is isoelastic with elasticity " < 1 and identical for all individuals. As labour input increases, rises at decreasing rates from zero to its maximum value, max : ¼ max :ðl H Þ" :. ð25Þ. With 1 ð1=LÞ and a ½1=ðLÞ þ ð1 Þ
(18) the first-order condition assumes the following form: " 1 H H a2 " þ 1 l zSt ð1 Þa l zSt 1 2 ¼. l H ð1 Þ : ð1 l H Þð1 Þ. ð26Þ. It can be shown that condition (26) has a unique solution if the resource stock per capita is small. This can be seen from the following intuition. As S becomes very small, the LHS approaches " þ 1, a constant. In contrast, the RHS rises from zero to infinity as l H increases from zero to one. It follows that a unique solution exists then. A formal proof is contained in Chakraborty (2001a). Furthermore, implicit differentiation of condition (26) reveals that dl H =d < 0 if the resource stock per capita is small. The first-best solution. Obviously, the first-order condition for is not affected by probability being a function of harvesting effort. It follows from (18) that the efficient insurance premium is ¼ 1. The first-order condition for l H can be expressed as 2 1 1 1 l H ð1 Þ H H l zSt "L " þ 1 l zSt ð1 Þ ¼ : ð27Þ L 2 L ð1 l H Þ By setting ¼ 1 in (26) and observing that ð1 Þ ¼ 1=L, it can be seen that the LHS of (26) and (27) are identical. However, the RHS of (27) is smaller than the RHS of (26). It follows that harvesting effort is higher under the first-best solution. 5.2. Yields are correlated For simplicity, it is assumed that variances and covariances are equal across fishing units, i.e. Varðci Þ ¼ Varðcj Þ8i, j and Covðci ; cj Þ ¼ Covðck ; cl Þ8i, j, k, l. With a f þ St litH ð1 Þ and b wð1 ljtH Þ þ St ljtH =L, income variance is then VarðIÞ ¼ Varðci Þ½a2 þ b2 ðL 1Þ
(19) þ Covðci ; cj Þf2abðL 1Þ þ b2 ½L2 3L þ 2
(20) g: We consider first the Cournot–Nash solution of Section 3.1.2, where f and b are treated as constants. With a0 @a=@liH , the first-order condition can be expressed as 1. wL Varðci Þ a Covðci ; cj Þ bðL 1Þ ¼ 0 : z z a. ð28Þ.
(21) 232. Rabindra Nath Chakraborty. Condition (28) is equivalent to condition (16) if individual harvests are independent, i.e. if Covðci ; cj Þ ¼ 0. Correlation across harvests causes the marginal certainty equivalent of income per marginal unit of expected net harvest income to be higher (lower) than under independence for any positive effort level if harvests are negatively (positively) correlated, which results in a higher (lower) level of harvesting effort. In terms of Figure 1, the straight line AA rotates upward (downward) if the covariance is negative (positive). The first-best solution. The derivative of the certainty equivalent of income with respect to is @ I ¼ ðSt l H Þ2 ð1 Þ½Varðci Þ þ Covðci ; cj Þ
(22) : @ If individual harvests are negatively correlated, it is efficient for harvesters to share all of their harvests because @ I=@ > 0. If harvests are positively correlated, the Cauchy–Schwarz inequality5 implies that Covðci ; cj Þ Varðci Þ. Hence, @ I=@ > 0 if Covðci ; cj Þ < Varðci Þ:. ð29Þ. The following analysis assumes that inequality (29) is satisfied. With being set equal to one, the other first-order condition @ I=@l H ¼ 0 can be expressed as 1. wL Varðci Þ a Covðci ; cj Þ bðL 1Þ ¼ : z z St. ð30Þ. The LHS of (30) is equal to the LHS of (28), which represents the Cournot– Nash solution. However, the RHS is smaller (as a0 ð ¼ 1Þ ¼ St =L in (28)), which causes equilibrium labour input l H to be higher in the first-best case. The sign of dl H =d. We consider the Cournot–Nash solution (28) and define a0 Varðci Þ aa0 Covðci ; cj Þ ba0 ðL 1Þ wL: z z It follows that @ ¼ ðSt Þ2 ð1 Þ½Varðci Þ ð1 Þ þ Covðci ; cj Þ
(23) z @l H S ð1 Þ : t ð1 l H Þ2. ð31Þ. @=@l H is negative if individual harvests are positively correlated. If harvests are negatively correlated, it can be shown (by using the Cauchy–Schwartz inequality and observing that z 1=Ll H ) that @=@l H is also negative if the resource stock per capita is small. The derivative of with respect 5. The inequality states that ½Covðci ; cj Þ
(24) 2 Varðci Þ Varðcj Þ. With Varðci Þ ¼ Varðcj Þ, the inequality implies jCovðci ; cj Þj Varðci Þ..
(25) Informal insurance in renewable resource harvesting. 233. to is @ ¼ St @. . 1 Varðci Þ l H St þ 2ð1 Þ z L 1 H Covðci ; cj Þ l St 1 2 : z . 1þ. ð32Þ. It can be seen by inspection of (32) that @=@ < 0 if the resource stock is small. Furthermore, it can be shown that @=@ < 0 if the resource stock per capita is small. Hence, a sufficient condition for dl H =d < 0 is that the resource stock per capita is small. Chakraborty (2001a) contains a formal proof.. 6. Conclusion The preceding analysis has demonstrated in a general equilibrium context that incentives exist among the users of a renewable resource to share part of their harvest if individuals are risk-averse. Sharing complements the smoothing of resource consumption that is achieved through the market purchase of resource units. However, the reduction in the variance of income that is achieved by sharing comes at a cost, as sharing induces a lower effort level in resource harvesting (moral hazard). As a consequence, it is (second-best) efficient for resource harvesters to share less than all of their income if the resource stock is small. This result is consistent with the empirical findings from Tonga (mentioned in Section 1), where producers share less than their entire income on both islands compared by Bender et al. (2002). More generally, incomplete risk-sharing can occur when the payment of the insurance premium is enforceable but the labour inputs required to produce the income flows to be shared are not. This result is an explanation of incomplete risk-sharing that is alternative to the imperfect enforceability of insurance payments analysed by Coate and Ravallion (1993). Furthermore, the moral hazard effect of sharing is reduced by the long-run increase in the expected value of the resource stock and, hence, of expected harvest. A higher level of sharing is consistent with a higher expected resource stock in the long run, which is again consistent with the Tonga case. The net impact on the certainty equivalent of income is ambiguous, however. As was shown by Chakraborty (2001b) for a ‘pure’ resource tax in a setting without uncertainty, the latter may outweigh the former, i.e. sharing may increase income per capita in the long run. Alternatively, it is possible that the longrun effect reduces the moral hazard effect without eliminating it. The effects discussed result from the interaction of two externalities. First, individual resource harvesters obviously do not consider the impact of their insurance payments on the incomes of all other resource users when making their labour allocation decision. Second, individual resource users do not take the impact of their labour allocation decisions on the equilibrium (expected) level of the resource stock into account when they decide on labour.
(26) 234. Rabindra Nath Chakraborty. inputs. The second externality is a common-pool externality that results from the assumption that the resource is exploited under open access.. Acknowledgements The results presented in this paper emerged from the research project ‘The Economics of Cultural Complements’ (O¨konomik kultureller Komplemente), which was funded by the Swiss National Science Foundation under grant 1214-055843. I thank the Foundation for its support. I thank Ernst Mohr, two unknown referees, and the participants of the 78th EAAE Seminar on ‘The Economics of Contracts in Agriculture and the Food Supply Chain’ and the Research Seminar at the University of St. Gallen for helpful comments on an earlier draft.. References Alkire, W. H. (1977). An Introduction to the Peoples and Cultures of Micronesia. Menlo Park: Cummings. Arrow, K. A. (1970). Essays in the Theory of Risk-bearing. Amsterdam: North-Holland. Bender, A. (2001). Fischer im Netz. Strategien der Ressourcennutzung und Konfliktbewa¨ltigung in Ha’apai, Tonga. Herbolzheim: Centaurus. Bender, A., Ka¨gi, W. and Mohr, E. (2002). Informal insurance and sustainable management of common-pool marine resources in Ha’apai, Tonga. Economic Development and Cultural Change 50. Besley, T. (1995). Nonmarket institutions for credit and risk sharing in low-income countries. Journal of Economic Perspectives 9: 115–127. Brander, J. A. and Taylor, M. S. (1998). The simple economics of Easter Island: a Ricardo– Malthus model of renewable resource use. American Economic Review 88: 119–138. Chakraborty, R. N. (2001a). Sharing as informal insurance in renewable resource harvesting. Institute for Economy and the Environment Discussion Paper 88. St. Gallen: University of St. Gallen, Institute for Economy and the Environment. http://www.iwoe. unisg.ch [Accessed 14.11.2001]. Chakraborty, R. N. (2001b). Could Easter Island have been saved? The impact of culture on renewable resource use in a Ricardo–Malthus framework. Institute for Economy and the Environment Discussion Paper 86. St. Gallen: University of St. Gallen, Institute for Economy and the Environment. http://www.iwoe.unisg.ch [Accessed 22.08.2001]. Coate, S. and Ravallion, M. (1993). Reciprocity without commitment: characterization and performance of informal insurance arrangements. Journal of Development Economics 40: 1–24. Hazel, P., Pomerada, C. and Valdes, A. (1986). Crop Insurance for Agricultural Development. Baltimore, MD: Johns Hopkins University Press. Ka¨gi, W. (2001). The Tragedy of the Commons revisited: sharing as a means to avoid environmental ruin. Institute for Economy and the Environment Discussion Paper 91. St. Gallen: University of St. Gallen, Institute for Economy and the Environment. http://www.iwoe.unisg.ch [Accessed 22.04.2002]. Ligon, E., Thomas, J. P. and Worrall, T. (2000). Mutual insurance, individual savings, and limited commitment. Review of Economic Dynamics 3: 216–246. Mauss, M. (1923/1954). The Gift. London: Cohen & West. Pratt, J. W. (1964). Risk aversion in the small and in the large. Econometrica 32: 122–136..
(27) Informal insurance in renewable resource harvesting. 235. Rosenzweig, M. (1988). Risk, implicit contracts and the family in rural areas of low-income countries. Economic Journal 98: 1148–1170. Ruddle, K. (1996). Boundary definition as a basic design principle of traditional fishery management systems in the Pacific islands. Geographische Zeitschrift 84: 94–102. Stiglitz, J. E. (1974). Incentives and risk sharing in sharecropping. Review of Economic Studies 41: 219–255. Stiglitz, J. E. (1990). Peer monitoring and credit markets. World Bank Economic Review 4: 351–366. Townsend, R. M. (1994). Risk and insurance in village India. Econometrica 9: 83–102. Townsend, R. M. (1995). Consumption insurance. An evaluation of risk-bearing systems in low-income economies. Journal of Economic Perspectives 9: 83–102. Udry, C. (1994). Risk and insurance in a rural credit market: an empirical investigation in Northern Nigeria. Review of Economic Studies 61: 495–526. Wiessner, P. W. (1977). Hxaro: a regional system of reciprocity for reducing risk among the !Kung San. Ph.D. Dissertation, University of Michigan, Ann Arbor. Woodburn, J. (1982). Egalitarian societies. Man 17: 431–451.. Appendix Proof of Proposition 1 The first part of the Proposition will be proved by contradiction. As was demonstrated in Section 3.1.2, constraint (15) is not satisfied for l H ¼ 0 _ l H ¼ 1, irrespective of the values of . It will subsequently be shown that I is not maximised by ð ¼ 1; 0 < l H < 1Þ _ ð ¼ 0; 0 < l H < 1Þ if the resource stock per capita is small. Case (a): ¼ 1, 0 < l H < 1. It follows from (23) that 1 ¼ 0. Inserting this result into (20)– (22) and observing that 2 0 yields the conditions @ I @g þ H ¼0 @l H @l @ I @g þ 0: @ @. ðA1Þ ðA2Þ. Solving (A1) for and inserting the result into (A2) gives @ I @ I @g=@ 0: @ @l H @g=@l H. ðA3Þ. It can be seen by inspection of (18) that @ I=@ ¼ 0 at ¼ 1. From (15), gð ¼ 1Þ can be written as gðliH ; ¼ 1Þ ¼ zwL þ ða=LÞ with a zSt z2 ð1 ÞSt ½ f þ ðSt liH =LÞ
(28) . @ I=@l H can be expressed as @ I=@l H ¼ zwL þ a. With g ¼ 0 and L > 1, @ I ¼ zwL þ zwL2 ¼ zwLðL 1Þ > 0: @l H From (15), @g=@l H can be expressed as @g=@l H ¼ z2 ð1 ÞðSt =LÞ2 < 0. Finally, @g=@ can be written as @g St H 1 þ 1 : ðA4Þ l 1 zð1 Þ ¼ zS t @ ¼ 1 L i If the resource stock per capita, St =L, is small, @g=@ < 0, which implies that the LHS of (A3) is negative, a contradiction..
(29) 236. Rabindra Nath Chakraborty. Case (b): ¼ 0, 0 < l H < 1. With ¼ 0, conditions (23) and (24) are satisfied only when 1 ¼ 2 ¼ 0. Inserting this result into (22) and (21) yields @ I @g þ H ¼0 @l H @l @ I @g þ 0: @ @. ðA5Þ ðA6Þ. Solving (A5) for and inserting the result into (A6) gives @ I @ I @g=@ 0: @ @l H @g=@l H. ðA7Þ. Computation of @ I=@l H and comparison with (15) reveal that @ I=@l H ¼ 0 for ¼ 0. Moreover, it can be seen by inspection of (18) that @ I=@ ¼ z2 ð1 ÞðSt l H Þ2 > 0 at ¼ 0. Finally, it can be verified from (15) that @g=@l H is non-zero at ¼ 0. This implies that the LHS of (A7) is positive, a contradiction. As I is not maximised at ð ¼ 1; 0 < l H < 1Þ _ ð ¼ 0; 0 < l H < 1Þ, it follows that the maximum is in the set ð0 < < 1; 0 < l H < 1Þ if the resource stock is small, which establishes the first part of the Proposition. To prove the second part, it is sufficient to show that @g=@j ¼ 1 > 0. Inserting (16) into (A4) yields @g l H ð1 ÞL 1 þ 1 : 1 1 ¼ zS t @ ¼ 1 ð1 l H Þ From (16) and Figure 1, l H ! 0 as S ! 1. This implies that, as S ! 1, the term in curly brackets approaches and @g=@ approaches þ1. It follows that the Kuhn–Tucker conditions are satisfied for the pair of values ð ¼ 1; l H ¼ l H Þ with l H given by the condition gðl H ; Þ ¼ 0 if S is high.. Corresponding author: Rabindra Nath Chakraborty, Institute for Economy and the Environment, Tigerbergstrasse 2, CH-9000 St. Gallen, Switzerland. E-mail: [email protected].
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