World Bank Reprint Series: Number 260 Pradeep K. Mitra A Theory of Interlinked Rural Transactions Reprinted with permission from Journal of Public Economics, vol 20 (1983), pp. 167-91. World Bank Reprints No. 225. George Psacharopoulos, "The Economics of Higher Education in Developing Countries," Comparative Educatiotn RevliewX No. 226. Katrine Anderson Saito and Delano P. Villanueva, "Transaction Costs of Credit to the Small-scale Sector in the Philippines," Economic Developmient anItd Cultutral Chlange No. 227. Johannes F. Linn, "The Costs of Urbanization in Developing Coun- tries," E0onottmic Dez)elopnment anid Cuiltural Chlanzge No. 228. Guy P. Pfeffermann, "Latin America and the Caribbean: Economic Performance and Policies," Southwtestern Rezview of Maniagemtienit anid Ecotnomiiics No. 229. Avishay Braverman and Joseph E. Stiglitz, "Sharecropping and the Interlinking of Agrarian Markets," A mericant Econaomic Revliew No. 230. 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Srinivasan, "General Equilibrium Theory, Project Evaluation, and Economic Development," The Theory and Experietnce of Economic Dev)elopment Journal of Public Economics 20 (1983) 167-191. North-Holland Publishing Coinpany A THEORY OF INTERLINKED RURAL TRANSACTIONS Pradeep K. MITRA* Development Research Department, The World Bank, PWashington, DC 20433, USA Received December 1980, revised version received February 1982 This paper argues that a landlordiowner-cultivator who is unable to monitor a tent'laborer's effort input will link a wage-cum-output sharing contract with the provision of consumption credit. The response of contractual parameters to (a) alternative opportunities available to tenant'laborers, (b) the cost of credit, and (c) the riskiness of cultivation is then examined. The analysis shows that public policy designed to alleviate rural poverty must recognize the delicate relationships between technological considerations (prohibitive monitoring costs) and modes of transaction (interlinked contracts) that arise in the absence of a complete set of markets. 1. Introduction Several writers on South Asian agriculture [Bharadwaj (1974), Bardhan and Rudra (1978), Jodha (1979)] have documented the widespread use of contracts which link labor, credit and output transactions among the same set of agents. Examples include the provision of credit by a landowner to tenants, wage laborers or farm servants and the leasing out of land to families relatively well endowed with nontraded factors such as bullocks and family labor. The rationale for the use of interlinked contracts may only be fully appreciated with reference to the wider social and political context within which the village economy is embedded. This paper, however, advances an economic explanation of the phenomenon which is rooted in the view that prevalent modes of transaction, of which interlinked contracts are an example, are shaped by technological considerations. Specifically, it is shown that interlinking is an efficient economic response to unequally distributed information arising from the uncertainty which characterizes subsistence agriculture.' This has the implication that a situation where interlinking by landlords has been abolished and tenant/laborers allowed unrestricted access to official credit markets on competitive terms can be Pareto-inferior for rural economic agents. The analytical framework set up to demonstrate these *An earlier version of this paper was presented at the ADC-ICRISAT Conference on Rural Labor Markets held in Hyderabad, India, in August 1979. Thanks are due to T.N. Srinivasan for many helpful conversations and Clive Bell and the referees for comments on earlier drafts. Views expressed are the author's and do not necessarily reflect those of the World Bank. 'A valuable account of the importance of uncertainty in agricultural decision-making is provided by Bliss and Stern (1981). 0047-2727/83,'$03.00 168 P.K. Mitra, Interlinked rural transactionis results is then used to examine the response of the contractual parameters to changes in (1) the returns to alternative occupations available to tenant/laborers, (2) the cost of credit to the village economy and (3) the riskiness of cultivation. The rest of this introduction attempts to explain the approach taken in the paper and to interpret the results reported above. The use of interlinked contracts is not easily explained in economies characterized by a complete set of markets. Tradiational subsistence agriculture is, however, subject to substantial risks which, in the presence of costly monitoring and moral hazard, preclude the functioning of such markets. Other institutions can then be expected to perform some of the roles economists usually assign to markets; it is argued below that interlinking is a response to such a need. The main point can be illustrated simply with an analogy. Costs of monitoring prevent a seller of car insurance from discovering to what extent an accident was due to bad luck or insufficient care. The probability of an accident can however be influenced by controlling transactions in related commodities so that insurers would like to see a tax on sales of alcoholic beverages to motorists. Likewise in agriculture. A landlord/owner-cultivator may be unable to tell whether the observed low output of a particular tenant/laborer is due to adverse circumstances or inadequate effort. There is, however, a conventional externality imposed by a cultivator's borrowing on the amount of effort he chooses to expend; efficiency in allocation then requires intervention in the credit decision. This takes the form of interlinking credit and output contracts. The above argument for interlinking is developed in the paper in the context of a principal-agent model. A group of identical tenant/laborers is brought together by a landlord/owner-cultivator who maximizes profits subject to tenants getting no less than a reservation utility level. Production decisions are influenced via a linear incentive system, i.e. one where y, the amount produced by a farmer, is related to z, the amount retained by him as follows: z=x+ fly; 7x,J>0. a and : may naturally be regarded, following Stiglitz (1974) as the 'pure wage' and output-sharing parameter, respectively. It can then be shown that Pareto efficiency requires exercising control over the amount of consumption credit made available to tenant/laborers. Three sets of comparative statics exercises are performed and their impact on the contractual parameters ascertained. The first traces the consequences of varying the reservation utility level summarizing alternative opportunities available to tenant/laborers. The second examines the effects of changing the cost of credit to the system. The third explores the results of a mean- preserving change in the riskiness of cultivation. The principal results for utility functions additively separable in present and future consumption and effort are given below. Let a denote the elasticity of a tenant/laborer's marginal utility of P.K. Mitra, Interlinked rural transactions 169 consumption. If the output-sharing parameter, /, is fixed at a conventional norm (50 percent in parts -of Indian agriculture), an increase in tenants' reservation utility level (1) cheapens the cost of credit provided by landlords, (2) reduces, leaves unchanged or increases the pure wage according to either a 1, and (3) causes more borrowing if a < 1. A mean-preserving reduction in the riskiness of agricultural production (1) raises the cost of consumption credit and (2) raises the pure wage if a> 1. If the intermittently discussed policy of abolishing interlinking were implemented and tenants given unrestricted access to official credit markets, subsidization of that credit would (1) lower the tenant's cropshare if r< 1; (2) lower the pure wage if a-> 1; and (3) reduce the amount of borrowing if a = 1. An important implication of the analysis developed in this paper is that public policy designed to help the rural poor must duly recognize the relationships between technological considerations (prohibitive monitoring costs) and modes of transaction (interlinked contracts) to which they give rise. Thus, well-meaning attempts to abolish moneylending by landlords and to grant tenants access to official credit markets at competitive rates would actually be Pareto-worsening for landlords and tenants. This result, which is rather striking, may be related to the line of work developed by Mirrlees (1974), Hart (1975), Diamond and Mirrlees (1978) and Newbery and Stiglitz (1979b). Those authors showed that the opening of a market which adds to an incomplete set of contingent markets, without however completing them, may generate Pareto-inferior outcomes. We close this introduction by mentioning related work on agrarian contracts. The principal-agent formulation of the landlord-tenant problem is developed with many interesting examples in Newbery and Stiglitz (1979a) and extended, most recently by Braverman and Stiglitz (1982), to analyze interlinked contracts. That model is used by Braverman and Srinivasan (1981) to examine sharecropping-cum-credit contracts in a village economy without uncertainty. Problems of coordination between landlord and money lender are also explored in Bell and Zusman (1980). The plan of this paper is as follows. Section 2 outlines the principal-agent model and derives the properties of a Pareto-efficient allocation both with and without moral hazard. It then demonstrates that a moral hazard constrained allocation is in general not attainable without interlinked contracts. Section 3 is devoted to comparative statics analysis. Section 4 concludes the papef. 2. Interlinked contracts 2.1. Farmers Consider a community of identical landless farmers/wage laborers who deal with a single landlord/employer. A typical farmer enters into a contract 170 P.K. Mitra, Interlinked rural transactions with the landlord/employer to work in period 1 in return for a wage, a, and a share, fi, of the output he produces. The harvest, however, becomes available only in period 2, so that the farmer must borrow to finance first period consumption. The loan is paid back, with interest, in the second period. Let -= consumption in period i (i = 1, 2), L =labour time, e =effort (O0, U3<0. It is also assumed that ui-*ox as cio-0 (i=1,2) and U3-X*- as e--l. Thus, the marginal utility of consumption and the marginal disutility of effort become infinite at zero consumption and maximum effort, respectively. An individual farmer's production possibility is subject to uncertainty- output, Y depends on a random variable, 0, representing the state of nature and on effective labour, e, which must be applied (in period 1) before the state of nature is known (in period 2): Y=f(e,Oh (1) where f is increasing and strictly concave in e, i.e. fe>0, f, <0. The variable 0 can assume a number of values Oj with known probabilities pj (j= 1, ...,n). To highlight the importance of uncertainty, it will be assumed that state of nature, k, where f'(e, Ok)=O, all t', may occur with nonzero probability. Relation (1) does not contain any explicit reference to land; its inclusion would not add much to the analysis presented here. 2.2. The lanidlord The profit accruing to a landlord in the jth state of nature equals the difference between production in that state of nature and the sum of wages and tenant/laborers' share of the crop, If the landlord has access to a perfect organized capital market at a rate of interest, i, his present value profit in the jth state of nature, Tji may be written T = r [(1- f(e, 0j)9- ] (j . 1), (2) where r = I (l + i). Since a and ft are agreed upon in advance of knowing which state of nature will occur, it is clear from (2) that Tj could be negative P.K. Mitra, Interlinked rural transactions 171 in states of nature where output f becomes zero. It is assumed that the landlord has other sources of income large enough to absorb those losses, Under these circumstances, the landlord is likely to be considerably less risk- averse than his tenant/laborers. We assume that he is risk-neutral and therefore that his objective is the maximization of expected profits, ET The landlord's task is to choose a risk-sharing scheme, here described by ac and ,B and hence restricted to be linear, and other control variables to specified to maximize expected profit rE[(l - ,3)f(e, 0)-ca] (3) subject to Eu[c, a + ,Bf(e, 0)- c(l + i), e] - u, (4) where (2) has been used in writing (3). Expression (4) states that the expected utility of a tenant/laborer must not fall short of d, a reservation utility level summarizing opportunities available elsewhere. Notice that second period consumption is the sum of wages and share of output less repayment of the consumption loan. It is assumed that tenants may borrow at the rate i as well; it will be shown that even with this assumption about access to the capital market on equal terms, efficiency in allocation in the presence of moral hazard will call for measures to influence a tenant's consumption loan decision. In the rest of the paper we shall formulate and analyze a sequence of models where the landlord can exercise progressively less control over individual farmers' decisions. 23. Unconstrained Pareto efficiency A useful benchmark for subsequent analysis is provided by an allocation where the landlord can directly control (a) the application of effort and (b) the amount of consumption in both periods, subject only to ensuring a tenant/laborer a utility level no smaller than a. The landlord's problenm is therefore one of choosing c, e, cc and /B to maximize (3) subject to (4). It will be established that the solution to this is Pareto efficient, i.e. that it does not permit an increase in ET (respectively Eu) without reducing or leaving unchanged Eu (respectively ET). To this end, we first provide an intuitive argument to show that (4) will hold as an equality at a solution to the landlord's problem. For if not, it is always possible to reduce the pure wage a slightly and keep c, e and /8 unchanged without violating the constraint (4). Since a feasible reduction in the pure wage increases the landlord's profit (3), the original situation could not have been optimal. We have therefore motivated JPE B 172 P.K. Mitra, Interlinked rural transactions Proposition 1. The reservation utility constraint (4) holds as an equality at a solution to the landlord's maximization of (3) subject to (4) with respect to the variables c, e, a and ,B. Proof. Since u2-*cox as second period consumption tends to zero, and there is a nonzero probability of states of nature where output is zero no matter how much effort is applied, aX must be positive at any solution to the maximization problem. The proof proceeds by contradiction. Suppose that (4) is a strict inequality at a solution to the landlord's maximization. Denote that solution by (c*, e*, a*, /*). Since * >0 and the utility function of tenants is continuous, there exists an arbitraril- small positive number, c, such that a= -a*- is positive and continues to satisfy (4). The effect of such a feasible reduction in cc is to increase expected profit in (3), i.e. ET(c*,e*,exo,/3*)>ET(c*,e*,o*,/,*). But this contradicts the supposed optimality of the starred allocation. Hence (4) must hold as an equality. The argument that (4) always holds with equality, together with the concavity of the utility function, implies that there exists a positive number, /., such that a solution to the landlord's problem (c*, e*, xc*, /3*) maximizes rE[(l - ,)f(e, 0)- ac] + AEu, (5) where A may be interpreted as the shadow price of (4) in terms of the landlord's expected profit. But this implies that the starred allocation is Pareto efficient. For if there were another feasible allocation (denoted by bars) with the property that ET Eu)-(ET*, Eu*) with strict inequality in at least one component, it would lead to a higher value of (5), a contradiction. This establishes Proposition 2. A solution to the expected profit maximization problem with respect to c, e, a, ,B, subject to a reservation utility constraint, is Pareto efficient. We next derive some other properties of a solution to (3) subject to (4). With (5) as the Lagrangean, the first-order conditions with respect to c, e, ac, ,B are Eu1 <-Eu2 (c - 0) (6) r (on remembering that r = 1/(1 + i)) r(1 -/)Efe + AE(u2/3fe + U3) < 0 (e > 0), (7) -r + AEu2 < 0 (a -0), (8) P.K. Mitra, Interlinked rural transactions 173 -rEf +±2Eu2f 0), (9) where each inequality bears the relation of complementary slackness with the variable in brackets appearing on the right. Since the landlord is risk neutral, and 'disasters' a real possibility, we should expect him to bear all the risk in any Pareto-efficient allocation. This leads to Proposition 3. A Pareto-efficient allocation is characterized by pure wage contracts alone, i.e. /B=0. Proof Let C2j =a farmer's second period consumption in the jth state of nature. The linear risk sharing scheme makes C2j = ct + /f(e, Of) - c(l + i), (10) where the pure wage component, a, is certain but /3f(e, Oj) is not. Suppose, contrary to the proposition, that /3>0. It then follows from (10) that cov [c2j,f(e, Oj)] >0, where cov denotes covariance. Let u2j =a farmer's marginal utility of consumption in the jth state of nature. By definition, U2j= u2(c, C2j)- Concavity of the utility function ensures that u2 is nonincreasing in c2j which, combined with cov[c2j,f(e,Oj)l>0 implies coV[u2,f(e, j)]<0. To summarize the argument so far, /3>0 implies cov [u2,f(e, 0j)] <0. (11) Since a>0 at a solution, (8) must hold as an equality. From (8) and (9) it then follows that Eu2f < Eu2Ef (3 - 0), i.e. by definition, cov [u2,f(e, Oj)] <0 (/3> 0). This relation implies that /3>0 implies cov [U2 ,f(e, )] = 0. (12) But (12) contradicts (11) so that /3=O at a Pareto-efficient allocation. 174 P.K. Mitra, Interlinked rural transactions The use of pure wage contracts implies from (10) that second period consumption, C2j=x-c(l +i), i.e. a constant independent of the state of nature. This result indicates that whether or not the landlord cultivates his land with pure wage labor is endogenous to the problem analyzed here. We therefore have Proposition 4. A Pareto-efficient allocation equates consumption across all states of nature. The complete insurance afforded a farmer under such an arrangement leaves him no incentive to work. Since effort confers disutility, the allocation is unattainable without the sort of centralized labor direction we have assumed. We next examine a second-best allocation where farmers are free to make their own effort supply decisions. 2.4. Second-best Pareto efficiency The second-best problem considered here arises because the landlord must respect the sovereignty of farmers' decentralized effort supply decisions. However, he continues as before to control their consumption decisions. This is in keeping with our strategy of analyzing models where the landlord exercises a progressively dirminishing degree of control. The consequences of allowing farmers to take their own effort supply and consumption decisions is central to the paper and will be examined at some length below. 2.5. Farmers Farmers, who are assumed to be expected utility maximizers, select their effort supply in period 1 before knowing which state of nature will occur; the state of nature is revealed to them in period 2. A farmer's problem is to choose e to maximize W=Eu[c, x+ff(e, 0)- c(l + i), e], which leads to the first-order condition We= /Eu2fe + Eu3 =0. (13) For future reference, it is useful to note that W,4ecPE[U21 -(1 + )U221fe+ E[u3l -(1 +i)u32], where the notation uij denotes cross partial derivatives. The effect of P.K. Mitra, Interlinked rural transactions 175 borrowing on effort is given by de WI'ec dc Wee Since W1, <0, by second-order conditions, de sign d = sign {fiE[U21 (1 + )U222if,e+E[U3 1-(1 + i)u32]}f (14) Similarly, WJ,4 3=EU22f + EU32 and sign d = sign f fEU22fe + EU32f. (15) dc 2.6. The landlord Monitoring costs are assumed prohibitive and preclude the landlord's ascertaining (a) the true state of nature on a farm both before and after the event and (b) the amount of effective labor (recall L = 1) applied by a farmer. He can only observe every farmer's output and is therefore unable to decide, for example, whether a case of low output is due to adverse circumstances or inadequate effort. This is a reasonable assurmiption: the pace, thoroughness, efficiency and investiveness of the agent, subsumed in the variable e, have been identified by Stiglitz (1974) and other writers on contractual relationships as being extremely expensive to monitor. Output therefore serves as an easily measurable but imperfect surrogate for what the landlord would really wish to know, namely effective labor input. The contractual parameters c, a and ,B, being based on what is observable by the landlord, are not state-dependent. Notice that the monitoring problem arises even if the landlord is dealing with one tenant/laborer. The assumption of a number of tenant/laborers who are identical ex ante but, because of the operation of the random element, nonidentical ex post does not affect the severity of the monitoring problem. The landlord's problem is to choose c, a. and ,B to maximize R=rE[( 1-)f fe(c, a,f/), 0}-r] (16) subject to V = Eu[c, a + 1f f e(c, ~, f), O} -c(1 + i), e(c, oc, fl)] - u. (17) 176 P.K. Mlitra, Irnterlinked rtural tratnsactions In order to show that a solution to the above problem is constrained Pareto efficient, it is necessary to argue as before that, under certain assumptiops, (17) will hold as an equality. We therefore establish Proposition The reservation utility constraint (17) holds as an equality at a solution to the landlord's maximization of' (16) subject to (17) with respect to the variables c, a and /3, provided that U432 <0 Proof. Denote a solution to the above problem by (c*,*./*). Suppose that (17) is a strict inequality at that solution. It has been established before that O*>0. Since the utility function and effort supply function are continuous, there exists an arbitrarily small positive number, i, such that CO= *-f is positive and continues to satisfy (17), its effect on Eu via the effort supply function e(c,oc,l/) may be ignored because of the envelope theorem. The effect of such a feasible reduction in a on expected profit is given, from (16), by R(c*, xo, 3*)-R(c*, *, /*) =r (I - f)gf, d je-| (18) where the derivatives appearing on the right-hand side of (18) are evaluated at (c*,x*,,B*). From (15), de,dx>O if u32<0 and R(c*,oc(,f*) is a superior feasible allocation, contradicting the supposed optimality of (c*, x*, /*). Hence, (17) must hold as an equality provided that the marginal utility of second period consumption does not increase with effort 2 An argument analogous to that preceding proposition 2 now establishes Proposition 6. A solution to the expected profit maxiiniZation probtlem with respect to c, a and /3 subject to a reservation utility floor is constrained Pareto- ejficient, where the constraint arises from the tenant's choice of efJbrt supply, provided 1u32<0. A constrained Pareto-efficient allocation is one where the landlord, although respecting maximizing farmers' effort supply decisions, continues to control the amount of consumption made available in the first period. This implies, of course, that individual farmers are not permitted to borrow and consume in accordance with their own wishes, a notably stringent requirement. We demonstrate that a second-best allocation is characterized by the landlord's prohibiting farmers unrestricted access to the organized credit market. The Lagrangean corresponding to the maximization of (16) subject to (17) 2A sufficiently rapid increase in the marginal utility of second period consumption with effort would lead to a large reduction in effort as x is decreased. rhis would prevent the landlord from using x as an instrument to push a tenant, laborer to his reservation utility level. P.K. Mitra, Interlinked rural transactions 177 is S/=rE[(1 - f)f {e(c,ca, f), 0} -ce] + 2(V-ui), where, from (17), V=Eu[c,ca+ 3f{e(c, a,/3),0}-c(1 +i),e(c,ca, f)]. On using the envelope theorem, the first-order condition with respect to c, the consumption loan size, is dc r(1-,Bl)d Efe+2I{=O. (18) Since A measures the increase in expected profit following a reduction in u, it is positive. Hence, VC50 according as d= " (19) A farmer, if allowed to choose c freely, would set Vc=O. From (19), this is satisfied at a second-best Pareto-efficient allocation only when de/dc=0, a condition that cannot generally be expected to hold. The landlord therefore uses the loan size as an instrument to influence a farmer's (unobservable) effort level. In words, (19) states that at a second best, a farmer would wish to borrow less (respectively, more) depending on whether effort increases (respectively, decreases) with borrowing. From (14) and (19),3 Vc 0 according as {/3E[u21-(1+i)u221fe + EEU31-(l + i)U32} O. (20) For initertemporally additive utility, i.e. for u = 0(c, e) + E[a + ,Bf -c(1 + i)], (20) reduces to Vc 0 according as c,e fl((l +i)EVi"fe, (21) where single and double primes denote the first and second derivatives of single-variable functions, respectively. We therefore have Proposition 7. With an intertemporally additive utility function, a tenant/laborer would wish to borrow less (respectively, more) than at a second- 3All derivatives are evaluated at the second-best solution. 178 P.K. Mitra, Interlinked rural transactions best Pareto-eficient allocation according to whether kce c (1 + Z)E/fe. Corollary. In the special case of additive separability (kce,=0), a tenant/laborer would wish to borrow less than at a second-best Pareto-efficient allocation. The above discussion shows that (constrained) Pareto efficiency requires that the consumption-leisure tradeoff be influenced by restricting farmers' access to the capital market. In this model, farmers may borrow at the same rate, i, as the landlord; it is nevertheless efflicient to control the amount borrowed at that rate directly. These results suggest that 'delinking' of credit contracts from output contracts would lead to a reduction in profits without any change in tenant welfare. This is formally recorded in Proposition 8. When the utility function is intertempurally additive and 0,ce-O, an interlinked credit-cum-output contract cannot be Pareto dominated by a delinked contract where farmers are permitted to make borrowing and effective labour decisions freely. Proof. A tenant/laborer's problem is one of jointly choosing c and e to maximize Eu[c, a± +f(e,0)-c(l +i),e]. It can be shown (see appendix 1) that de/dic<0 provided ckce-O. The landlord's task is to choose a and ,B to maximize rE[(I - #)f e(b, /), ' } - ] (22) subject to Eu[c(ox, /3), ox + 3f {e(ox, /3), 0x-c(x, ,B)(1 + i), e(a, /3)] - u. (23) An argument similar to that used to establish proposition 5 [with the envelope theorem applying to both c(c,/) and e(x,,B)] then shows that (23) will hold as an equality. A tenant/laborer is thus no better off than at a second-best Pareto-efficient allocation. The maximum value of (22) cannot be greater and will in general be less than that attainable in the second-best problem. This is because the landlord must make do with controlling a and f/ rather than the triple (x, ,/, c). This establishes proposition 8. P.K. Mitra, Interlinked rural transactions 179 We have therefore provided an economic argument which helps explain why landlords can impose significant restrictions on farmers' freedom to operate on other markets. Such restrictions can then be sanctioned by social custom and implemented through the use of extra economic coercion. But to help appreciate the economics of the argument better, consider a somewhat different example. Suppose that a group of landless farmers is brought together by a cooperative authority which arranges production and distribution to maximize the collective welfare of its constituents, subject to the need to pay landowners a minimum rent for leasing out land. Since this formulation of the problem interchanges the objective and constraint in (16) and (17), it follows from proposition 7 that the cooperative authority will link credit and output contracts in the interest of its own members. The argument for interlinking is therefore based on efficiency and not on the institutional circumstances which assign principal and agent roles to particular individuals. This has the important implication that public policy designed to help the rural poor must recognize the relationships between technological considerations (prohibitive monitoring costs) and modes of transaction (interlinked contracts) that arise in the absence of a complete set of markets. 3. Comparative statics This section specializes the model to ascertain Ihe response of contractual parameters to changes in underlying data describing the economy.4 Specifically, it is assumed (1) that there are two states of nature, 01 and 02, occurring with probabilities p and (1 -p), respectively; (2) that effort e can take two values, 0 and 1 (as before L= 1); (3) that the production function y = f(e, 0) has the following properties: f(01, 1)=Q 1 f(02, e) = 0, for all e ; (24) f(0,0)=0, for all J Thus the cause of zero output can either be bad luck or lack of effort; (4) for any pair (c1, c2), the utility function where e = 1 (u'(cl, c2)) and where e=0 (U0(C1,C2)) are related as follows: u'(cl, c2) = U°(c1, c2)-A, (25) where A>0. 4The model is an adaptation of that introduced by Diamond and Mirrlees (1978) to analyze social insurance. 180 P.K. Mitra, Interlinked rural transactions The second-best Pareto-efficient allocation with control over consumption credit and an allocation with completely unrestricted access to organized credit may both be regarded as polar cases which serve to organize our thoughts on this subject. An intermediate formulation which allows the landlord partial control is to suppose that he can set a price at which agents undertake utility-maximizing borrowing. Let t denote an interest tax or subsidy charged to agents, so that ti is the rate of interest at which they may borrow in the first period. If the utility function is assumed intertemporally additive, the expected utility from applying effort is 0(c')+pl[a+flQ-c'(1 +ti)] +(1 -p)O[a-c'(1 +ti)] -A, (26) while the expected utility from not applying effort is OMc° + 0[a -co(' + ti)], (271) where c' = optimal borrowing when e =1, and co optimal borrowing when e=O.5 To make farmers choose e=1, it is necessary for the landlord to ensure that (26) is at least as large as (27). To simplify the algebra it will be assumed that borrowing decisions are made on the premise that e = 1, so that the constraint may be written 0(c)+p4[a{+flQ-c(l +ti)] +(I -p)O[,x-c(l +ti)] -A > o(c) + 0[a -c(l + ti)], where c is optimal borrowing when e= 1. This may be simplified to A KU + 3Q - cl +ti)] - O[C- c(l + ti3] - - - . (28) p Expression (28) is sometimes referred to as the moral hazard constraint. With these assumptions, the landlord's problem is one of choosing a, / and t to maximize F= 1 i p(l - #)Q -a - OX, , 00 ) 1-) (29) 1 subject to H = ±[a + BQ -c(a, /3, t)(1 + ti)] - /[a-c((a, , t)(1 + ti)] -A> , (30) p 5This formulation assumes that farmers do not default on their consumption loans when it suits them to do so. P.K. Mitra, Interlinked rural transactions 181 R = 4(c(a, 3, t)) + pOEc + 3Q-c(x, ,B, t)(1 + ti)] +(1-p)0[aX-c(a, 9, t)(l + ti)]- -1Z O, (31) where (30) is the moral hazard constraint and (31) ensures that expected utility when effort is applied is no smaller than the reservation utility level, a. To derive comparative statics results, form the Lagrangean M=F+rqH+±R. (32) Reservation utility. We first consider how the variables a, : and t vary with u, which summarizes alternative opportunities available to tenant/laborers. This calls for a solution to the following system: Maor Map M.t H. R2 da -MUU Mp,a Mpg Mpt Hp Rp d,B -Mp M.a M,P Mt H, R, dt - -M," du. (33) Ha Hp H, 0 0 dq -Hu Ra Rp Rt 0 0 d7r -Ru Solutions to the complete system (33) are quite complicated. We have therefore found it useful to concentrate on a number of special cases derived by holding constant one of the variables /, t and a. The propositions which follow in this section arise from straightforward but tedious manipulation of equation systems like (33); proofs are sketched in appendix 2. (1) Constant ,B. Imagine a society where the cropshare /3 is conventionally fixed (e.g. at 50 percent in some parts of India). On deleting the ,B row and column in (33), we are able to derive Proposition 9. With /3 held constant: (i) dt/dui<0, (ii) da/di' 0 as u 1, and (iii) dc/du > 0 if a - 1. An increase in reservation utility causes landlords to cheapen consumption credit [part (i)]. A sufficiently rapid decline in the marginal utility of consumption (u > 1), however, requires a cut in the pure wage ca to maintain work incentives [part (ii)]. Borrowing unambiguously increases in response to cheaper credit if G < 1 [part (iii)] in which case the pure wage goes up as well [part (ii)]. (2) Constant t. A special case of this is the situation where the institution of interlinking is successfully abolished and tenant/laborers are granted access to official credit at competitive rates of interest. The next proposition is however not confined to the case where t = 1. 182 P.K. Mitra, Interlinked rural transactions Proposition 10. With t held constant: (i) d,B/dtu> 0, (ii) doc/dzi> 0, and (iii) dc/du-> O.. These results are intuitively satisfactory and require no explanation. (3) Constant a. With an institutionally fixed pure wage, we are able to derive Proposition 1]. With or held constant: (i) df,/dtu>O if a- 1; (ii) dt/duz 1; and (iii) dc/du->0 if > 1. An increase in reservation utility is accompanied by measures to preserve incentives - increasing agents' cropshare and cheapening the cost of credit - when the marginal utility of consumption declines rapidly. Cost of credit. We next examine the consequences of changes in the cost of organized credit, i. The system to be solved is M=: M.ap M., Ht R. da -- Mai Mfa M , a t Hp Ra dc -Mpi MtK Mt# Mt, Hi RI dt = -M1i di. (34) Ha Hp Ht 0 j d1j -Hi Ra, R# Rt O O d7r --Rj We may now establish Proposition 12. With t held constant: (i) dp//di>O if u<1; (ii) doc/di>O if o-1; and (iii) dc/di>0 if 0= 1. This proposition shows that in a situation characterized by no interlinking, a cheapening of official rural credit will lead landlords to reduce the tenant,'laborer's cropshare when o< 1. It leads to a cut in the pure wage in the case where maintenance of work incentives is potentially important (ca> 1). The combination of the above leads to the following seemingly perverse result when u= 1. Borrowing for consumption decreases when official credit is cheapened because of offsetting actions by landlords. Riskiness of cultivation. We conclude this investigation by examining the effects of a mean-preserving change in the riskiness of cultivation. Since mean output equals pQ, it is postulated that p and Q change together while satisfying d[pQ] =0, i.e. pdQ+Qdp=O. (35) P.K. Mitra, Interlinked rural transactionis 183 The variance of the distribution is p(l-p)Q2 and d[p(1-p)Q21/dp= -Q2<0 when d[pQ]=O. Hence, dp>(O) satisfying (35) corresponds to a mean-preserving reduction (increase) in risk. The effects of such a change may be determined by solving the following system: Mal Map Mat Ha R da = Map Mpa Mpp M0t Hp Rp d,B -MpI Mta Mtp Mtt H( Rt dt = -M,p dp, (36) Ha Hp H, ° ° dn -Hp R, Rp R, 0 0 d7i -RpJ where dp and dQ are related as in (35). Proposition 13. If the moral hazard constraint (30) is binding, then withfixed ,B: (i) dt/dp > 0 and (ii) do/dp > 0 if o- 1, provided Y" > 0, where three primes denote a third derivative. The assumption of /" >0 is plausible and is implied by the hypothesis of decreasing absolute risk aversion. With (30) binding and ,B constant, a mean- preserving reduction in risk increases the utility of working relative to that from inactivity. This reduces the magnitude of the externality to which interlinking is a response. Recall that with intertemporally additive utility, agents wish to borrow less than the principal would like, thus establishing a presumption for subsidized credit.6 A reduction of interlinking is then associated with a mean-preserving reduction in risk [part (i)]. The need to preserve work incentives when au 1 leads to an increase in the pure wage Epart (ii)]. It is worth noting that borrowing may go up or down depending on the particular form of the utility function. Furthermore, the landlord's profit, F, as defined in (29), may either increase or decrease following a mean- preserving change in risk. Thus, while a mean-preserving reduction in risk is a particularly simple characterization of technical change, it serves to illustrate that, under certain circumstances, a landlord will resist adopting a technical innovation. This arises because of the second-best nature of the problem and the need to devise incentive schemes which balance risk-sharing with the preservation of work incentives. 6The phenomenon of subsidized consumption credit was widely observed by Bardhan and Rudra (1978) in a sample of 110 villages in West Bengal. 184 P.K. Mitra, Interlinked rural transactions 4. Conclusion This paper has argued that the interlinking of labor, output and credit contracts often observed in rural economies can be regarded as an attempt to improve allocative efficiency in the face of moral hazard. It was shown that all Pareto-efficient allocations (regardless of their particular distributional characteristics) require a combination of wage-cum-output sharing with consumption credit contracts. The response of such contractual parameters to changes in underlying economic circumstances was also explored. Further work could usefully focus (a) on extending these methods to treat the case of ex ante heterogeneous tenants, and (b) on more dynamic situations where landlords and tenants have the opportunity of revising contracts in the light of reputations acquired in previous periods.7 Finally, there is the very interesting question of what policy conclusions are valid in a world where a rich panoply of rural institutions substitutes for the economist's landmark of a complete set of markets. We hope to address some of these issues in a subsequent paper. Appendix 1 This appendix establishes that when agents with intertemporally additive utility functions are free to choose consumption and effort, de/da O. This fact is used in the proof of proposition 8. An agent's problem is to choose c and e to maximize W= 4(c, e) + E# [ + /f (e, 0) -c(l + i)]. This leads to WC = Oc- (1 + i)Eqlc = °, We =be + lEfcfe = 0, WCC-Occ+(1 +i)2E#c° Wee = dee + ±PE[1cc± +lcfee] < 0, Wce = Ocef-i(1 + i)EVccfe To ascertain the effect of changing x on consumption and effort, it is 'An interesting recent paper by Radner (1981) suggests that repeated games between principals and agents allow the former to surmount the monitoring problem under fairly stringent conditions. P.K. Mitra, Interlinked rural transactions 185 necessary to solve the following system: WF W ce]F Fdc W d( Ldco~eL]-ch, (A.l1) W,4 =-( 1 + i)ENcc > 0, W,,. = E@ccfe < 0. From (A. 1): de I Wcc-WC.{ dox DI W4c-Wea' wnere D is the determinant corresponding to the matrix on the left-hand side of (A.1). From the second-order conditions, D>0. It is readily checked that de/da < O if (ce°0. Appendix 2 This appendix briefly sketches the manipulations underlying the propositions established in section 3 of the text. Agents' problem: An agent who sets e = 1 chooses c to maximize W=O(c)+ pO[1+#Q -c(l +ti)] +(1 -P)OEO-c(l. +ti)]. (A.2) This leads to: WC = '- (1 + ti)[pt'l + - p)0,0]= 0, WC0 + (I + ti)ftpol + (1 _ p)011] < Os WC. = - + ti)[pO'l +(1 - p)0,0, >O0, Wcfl= -(I1 ti)po' Q > , Wct= - i[pYl +(1 -p)q$0 +ci(l + ti)[pb' +(I -p)O] <0, Wa = -t[p44 +(1 -p)050 +±ct(l +ti)[p1 +±(1 -p)qO5] <0. With d(pQ)=0, 186 P.K. Mitra, Interlinked rural transactions WC,=-(1+ti)[(0'1-0-0)-f/Q4'1]>0 if O' is convex because of the convex function inequality (O' - O') < Qob. From the above, dc Wc. (1 +ti)[pOY+ (1-P)WOj 0 da wcc WCC dc Wcp (1 + ti)pc1Q >0 d,B Wcc wcc dc WcJ if[pq'4 +(1 -p)00] -c(1 + ti)[pb" +±(1 -p)O <]} 0 dt wcc wcc Note that dc dc i[po' +(t-p)0,] <0 dc Wci t{[pO'l + (1 -p)00 -c(1 + ti)[pc'f + (1 -p)'O]} <0. di Wcc Wcc Substituting from above for Wcc, using the first-order conditions Wc=O and remembering that o= -c"/l/, we may write dc, ct+(1±+ti) 0 according as a' 1. di With d(pQ)=0, dc Wcp (I +ti)[W(-0l-¢O) - NO1 > 0 dp wcc wcc Principal's problem Choose cc, ,B and t to maximize F = l p(l - f)Q - - C(a, ,S, t)(- 1-] 1+ subject to A p P.K. Mitra, Interlinked rural transactions 187 R = O(c(a, ,B, t)) + p( [ac + ,Q - c(a, /3, t)(1 + ti)] + (1 -p)OEx- c(a, lS,t)(l + ti)] -u1>O0, where c(a, /B, t) solves (A.2) The relevant derivatives for comparative statics are H.(01, - 0'OW < 0 wcc H (1 + ti)20,1pQ(O wcc H- = ic --l)H,:2 0 as a Hi (P1'- OLct+ ( + ti, di] as o-21. With d(pQ)=0, Hp=-(--4Q 1j-(1+ti)(0Y1-00')y- If H> 0 holds as an equality, i.e. A - 01 -001 p it follows that A p-- NC= (OI- 00 - ,Qoll) > p because 0 is concave and Hp >O, H,= 0, R. = p 1 + (1 -p)' > 0, 188 P.K. Mitra, Interlinked rural transactions Rp =pOQ$'Q>O, R1 = -icRa < 0 Ri =- etR, < 0, Rp=(0j-0o)-/3Qb' 0, by concavity of q, R=- 1. Proof of proposition 9. With ,B constant, eq. (33) of the text becomes Fmxa M=, Ha Ra d mau Mta M,, Ht Ri dt -MtiidU_A3 Ha Ht 0 ° d = ° Ra R, O O d7z l The determinant, D, corresponding to the matrix on the left-hand side of (A.3) is positive, by second-order conditions. Consider the expression 1 Hi= HaRt-HtRR ic =-ffH,R,>O. It follows from Cramer's rule that: (i) dt/dz7=(1/D)Haij <0; (ii) dca/dii= --H,B, 0 according as u1; and (iii) since c = c(a, /3, t) and /3 is fixed, dc _ dc dax dc dt da dx da dt d f Proof of proposition 10. With t constant, (33) becomes Maa Map Ha R dxc M.U M2a Mpp Hp Rp d/ _ -Mp d Ha Hp O O d =d -Ra Rp O O --I dc __1__ -- P.K. Mitra, Interlinked rural transactions 189 The relevant determinant, D, is again negative. Consider the expression B2 = HaRp-HPRa < °. From Cramer's rule: (i) d di7 D (ii) dc~ 1=D(H)2>O dc dc da dc d,B di(i da dt7 df dti Proof of proposition 11. With x constant, (33) becomes Maa Mpt Hp Ra dfl - Mp M,t M,, Ht Rt dt -Mtd Hp H, O O dq - 0 u Ral R, O O dr 1dJ Notice that B3=H#R,-H,Rp Q if a 1. Proof of proposition 12. With t constant, (34) becomes MXX Mmp Ha, R,] d1 --M,, mpg Maa Hp Ra df, - diM HL Ha 0 ° di -Hi R., Ra ° ° d7r -Ri 190 P.K. Mitra, Interlinked rural transactions From Cramer's rule: (i) d: --- (HiR- HaRi)B2 > 0, if uf<- 1, di D a_ (ii) di = -D(H0Rj-HiRp)B2 >0, if > 1, dc dc da dc d,B I di da di +d1 di Proof of proposition 13. With fl constant. 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