Peasants' Risk Aversion and the Choice of Marketing Intermediaries and Contracts: A Bargaining Theory of Equilibrium Marketing Contracts
Peasants' Risk Aversion and the Choice of Marketing Intermediaries and Contracts: A Bargaining Theory of Equilibrium Marketing Contracts
Abstract and Keywords
This chapter looks upon the nexus of contracts in the village marketing system as a bilateral bargaining game and explores the properties of the equilibrium set of contracts.
Keywords: bilateral bargaining, commission agent, contracts, equilibrium, marketing intermediaries
1. Introduction
Agricultural production is geographically dispersed, and individual producers often enjoy only limited direct access to central auction markets. This is particularly true in rural economies in which the communication and transportation systems are underdeveloped. Furthermore, in most economic systems marketing constitutes a specialized activity, which is carried out by specialized marketing intermediaries. The marketing system through which farm products flow from the original producers to the final consumers consists of the various intermediaries and market facilities as well as the contractual arrangements that mediate exchange between producers, marketing intermediaries and final consumers. These components of the marketing system constitute agrarian institutions of crucial importance.
There exists, at present, a fairly extensive literature dealing with marketing systems, including agrarian ones. Interested readers are referred to the comprehensive review by Wood and Vitell (1986). The main objective of this chapter is to offer a rigorous theory of a particular marketing system which, hopefully, may be extended to a wide variety of rural economic institutions.
The present study is concerned with marketing at the village level. At this level the number of marketing intermediaries and producers is of necessity small. Consequently market imperfections arise; exchange tends to become personalized, and competitive (‘auction’) markets are supplanted by various forms of bilateral contractual arrangements between producers and marketing intermediaries.
Marketing contracts are often simple transactions in which the title to the good is transferred at a given fixed price at a particular moment in time. In other cases, however, contracts may involve more complex arrangements in which the parties' payoffs depend on their actions and on the realized state of (p.298) the world. In view of the parties aversion to risk and the prevailing informational asymmetries between agents, marketing contracts are likely to include risk‐sharing and incentive provisions in addition to exchange price‐like parameters.
In the following sections I shall advance a static theory of marketing contracts which is designed to explain the determination of contractual terms in marketing contracts and the allocation of product to the various distribution channels under equilibrium conditions. The theory portrays the marketing system as a nexus of bilateral contracts among individual actors. The contracts involve agency relations;^{1} each agent may deal with several principals and each principal with several agents. As actors are diverse, contracts are idiosyncratic and are reached through bilateral bargaining. The theory draws upon the bargaining theory due to Nash (1953). As contracts are interdependent in many respects, we are interested in the equilibrium set of all contracts in the market — an equilibrium characterized by Zusman and Bell (1989). An earlier application of the contract‐theoretic approach to the theory of marketing channels was attempted by Zusman and Etgar (1981).
Although the theory is rather general, it is best illustrated with reference to a concrete example. In particular, we shall consider a simplified agricultural marketing system not unlike those found in developing agrarian economies. Accordingly, I first present a model of a village‐level marketing system; second, I characterize the equilibrium set of marketing contracts that make up the system and derive the equilibrium conditions; third, the equilibrium values of the contractual terms are studied analytically and numerically; and finally, the economic efficiency of the contractual system is considered.
2. The Marketing System
The market for paddy at the village level in India will provide the model used in presenting the present theory. Several types of marketing intermediaries may be found at the village level. Thus, according to K. Subbarao, who studied rice marketing in Andhra Pradesh in India, ‘The three principal agents who procure paddy stocks for delivery at the mill site are: (a) local commission agents (‘brokers’); (b) rice millers’ agents, and (c) cooperative processing societies’ (Subbarao 1978; 15). I shall adopt this description as my model, but in order to simplify the analysis shall consider only the first two marketing agencies.^{2} Furthermore, I shall assume the marketing system to consist of a single miller's agent, a single broker, and n producers whose output consists of a single homogeneous product. The functional characteristics of the various transactors are as follows.
2.1. The Miller's Agent (The Miller)
The miller purchases the product from the ith producer at the fixed price P _{i}. (p.299) The value of the product at the mill site is Z, and the handling cost is C _{M}(Q ^{M}), where Q ^{M} is the total amount of the product procured by the miller. C _{M}(Q ^{M}) is assumed to involve increasing marginal cost. Let Q _{i} be the total amount of product marketed by the ith producer and q _{i} the amount marketed by him through the broker, so that Q _{i} − q _{i} units of product are sold to the miller; then
The contract between the ith producer and the miller consists of two contractual parameters, P _{i} and Q _{i} − q _{i}.
2.2. The Commission Agent (‘Broker’)
The broker performs two functions: he seeks high‐paying buyers, and he handles certain phases of the market logistics for the producer. The search for buyers consists of sampling s potential buyers and entering into a transaction with the one offering the highest price, P _{s}. The search cost is assumed to be proportional to the number of buyers sampled. The broker's handling cost C _{B}(Q ^{B}) also involves increasing marginal cost. The total amount of product marketed by the broker is
(p.300) The contract between the broker and the ith producer includes two contractual parameters, q _{i} and v _{i}. Since individual producers are ordinarily not in a position to monitor the search behavior of the broker, the choice of the sample size s is the broker's discretionary action, and is not spelled out in the contract. Given his contracts with all producers, the risk‐neutral broker chooses s so as to maximize his expected income. The first‐order condition for maximum E(Y ^{B}) is, therefore,
The second‐order condition for maximum requires that $\partial {\overline{P}}_{s}/\partial s$ be decreasing in s (i.e., $\partial {}^{2}{\overline{P}}_{s}/\partial {s}^{2}<0$), and, since ${\overline{P}}_{s}$ increases in s,
That is, an increase in the quantity marketed by any producer through the broker, or an increase in the broker's commission fee, will encourage him to intensify his search effort—the incentive effect of the commission fee. This will benefit all producers availing themselves of the broker's services. The broker's discretionary decisions concerning information‐gathering, therefore, involve certain externalities. The effects of q _{i} and v _{i} on the search effort, and thereby on the broker's and producer i's expected utilities, are taken into account in the contract between them. However, the associated effect on the other producers is ignored.
2.3. The Producer
Producer i has Q _{i} units of product, of which q _{i} units $(0\leqq {q}_{i}\leqq {Q}_{i})$ are marketed through the broker and Q _{i} − q _{i} units are sold to the miller. His income is, therefore,
The producer preferences are given by his expected utility, E{U _{i}(Y _{i})}, where U _{i} is a von Neumann concave utility function. That is, the producer is assumed to be risk‐averse.
(p.301) 3. The Equilibrium Set of Contracts
As indicated earlier, each producer may enter two marketing contracts: one with the miller involving the contractual parameters Q _{i} − q _{i} and P _{i}, and one with the broker involving the contractual parameters q _{i} and v _{i}. How are these contracts determined?
In the present model contracts are determined through simultaneous bilateral bargaining between transactors, and the resulting equilibrium set of contracts represents the parties' market power.
The solution concept employed in this analysis adopts Nash theory of a two‐person bargaining game (Nash 1953) in deriving the contractual values of P _{i} and v _{i} in the corresponding contracts, given the product allocation among intermediaries and the terms of all other contracts.^{3} Notice that, while q _{i} and Q _{i} − q _{i} are both contractual parameters, it is impossible to determine their values by solving either the producer–miller contract or the producer–broker contract separately. This is because, being allocational contractual parameters, they have to satisfy the product availability constraints and must, therefore, be handled simultaneously. Hence, the allocation of product among marketing intermediaries is obtained by solving for the entire equilibrium set of contracts. However, an additional definition has to be introduced before undertaking a full characterization of the equilibrium set of contracts.
A reallocation of product between marketing intermediaries will be called an acceptable reallocation if it is associated with compensatory changes in the contractual parameters, P _{i} or v _{i}, so as to leave the expected utility of the relevant intermediaries unaltered. The equilibrium set of contracts will now be characterized by the following three conditions:

1. Given all other contracts and the product allocation, the agreed price P _{i} in the contract between the miller and producer i is a solution to a Nash bargaining game.

2. Given all other contracts and the product allocation, the agreed commission fee v _{i} in the contract between the broker and producer i is a solution to a Nash bargaining game.

3. No acceptable reallocation of product which may raise the expected utility of any producer exists.
Conditions 1 and 2 in effect imply that, given the product allocation, all contracts must be mutually consistent solutions to the corresponding bilateral bargaining games. In particular, if producer i and the broker disagree, the producer's conflict strategy is to sell his entire crop Q _{i} to the miller at the agreed price P _{i}. Similarly, if the miller and producer i fail to reach a contract, the producer will sell his entire crop through the broker at the (p.302) agreed fee v _{i}.^{4} Hence, the other given contracts determine the disagreement payoffs of the parties, and thereby their bargaining power. In other words, if any contract in the equilibrium set of contracts expires, then, given the remaining contracts in the set, the renegotiated contract will be identical to the one just expired.
Condition 3 also implies that in equilibrium no producer can renegotiate a new set of contracts (including a reallocation of his product) more advantageous to him which is acceptable to both marketing intermediaries.
Now, it has been shown by Zusman and Bell (forthcoming) that the equilibrium set of contracts corresponds to a solution of a non‐cooperative Nash–Cournot game, where for each i the ‘objective function’,
4. Determination of the Contractual Parameters
The contractual parameters may now be determined by maximizing W _{i} in (9) with respect to P _{i}, v _{i}, and q _{i} subject to the broker's behaviour (5), the product availability constraint, ${Q}_{i}{q}_{i}\geqq 0$, and the non‐negativity requirements ${P}_{i}\geqq 0\text{,}{v}_{i}\geqq 0$, and ${q}_{i}\geqq 0$. The W _{i} are maximized for all producers. Forming the Lagrangean expression
Here, μ_{i} may be interpreted as the shadow price associated with the product availability constraint of the ith producer. Adopting the utility normalization rule that at the solution point E(U′_{i}) = 1, we obtain from (18a) and (18b) that for P _{i} > 0 we must have λ_{Mi} = 1, so that from (12), (13), and (15) we have
A closed‐form solution of P _{i} may be obtained from (22) by using the approximations
Recalling that λ_{Mi} = 1 and E(U′_{i}) = 1, and using (6), (27), and (28), we get from (20a)
We shall now state two propositions concerning the allocation of product among the marketing intermediaries.
Proof. By (23),
PROPOSITION 2. If
Proof. We shall show that, under (33) and q _{i} = Q _{i}, conditions (18a)–(21b) are met. Since q _{i} = Q _{i}, (18a), (18b), (21a), and (21b) are clearly satisfied, while (19a) and (19b) are satisfied by the choice of ${v}_{i}^{*}$. By (27) and (28), we have
According to Proposition 1, a producer will sell his entire output to the miller if the expected highest price net of marginal costs that the broker can fetch in the market (given that this producer declines to use the broker's services) is smaller than the marginal net value of the product to the miller. According to Proposition 2, a producer will market his entire product through the broker if the net expected highest price that the broker can fetch (given that the producer markets his entire output through the broker) exceeds the net value of the output to the miller. Notice that the net broker's price is obtained by subtracting the marginal handling costs, C′_{B}, and the ‘marginal risk premium’, ${R}_{i}(1{v}_{i}^{*}){Q}_{i}{\sigma}_{{s}^{*}\text{,}i}^{2}$, from ${\overline{P}}_{{s}^{*}\text{,}i}$. Hence the more risk‐averse is the producer, the less likely he is to market his entire crop through the broker. It also follows from Proposition 2 that, if any risk‐averse producer (producer i, say) markets his entire output through the broker, then so do all risk‐neutral producers (R _{i} = 0) whose outputs, Q _{i}, are not greater than Q _{i}.
Propositions 1 and 2 leave a certain range for mixed choices of marketing intermediaries. In general, one expects risk‐neutral producers to prefer the broker, low and medium risk‐averse producers to allocate output among both channels, and high risk‐averse producers to prefer the miller.
Finally, it is worth noting that the increasing marginal cost assumption enhances the likelihood of intermediaries’ coexistence.
5. The Case of Numerous Producers
How are the contractual parameters, and the associated distribution of gains, influenced by the number of producers, n? Evidently, changes in n are bound to affect the structure of bargaining power and thereby the contracts’ terms. In order to facilitate the analysis, we shall adopt the simplifying assumption that all producers are identical in all respects; namely, they are all endowed with equal amounts of marketable product, and share the same attitude towards risk. Furthermore, since we are interested in the effects of variation in n, and not in output, we shall also assume that the total amount of output in the village economy is constant independently of n. Finally, it is assumed that both distribution channels are used by producers. In view of our increasing‐costs assumption, this is rather natural. Symmetry considerations then suggest that, for all i, the following hold:
(p.308) From equations (11), (23), and (26), we then get, for all i,
PROPOSITION 3. In the equilibrium set of marketing contracts with numerous identical producers, the expected absolute value of the broker's fee per unit product, $v{\overline{P}}_{s}$, is equal to his marginal handling cost, C′_{B}; and the price, P, paid to producers by the miller is equal to the value of the product at the mill site, Z, net of the miller's marginal handling cost, C′_{M}.
The marketing intermediaries’ price–cost relationships when producers are identical and numerous are depicted in Figures 15.1 and 15.2. The curves (p.309)
6. A Numerical Illustration
In order to gain further insight into the determination of the equilibrium set of contracts and its response to environmental and structural changes, a numerical analysis of a particular marketing system was undertaken. The algorithm employed in deriving the equilibrium solutions was based on the Nash uncooperative game representation of the equilibrium set of contracts as derived by Zusman and Bell (1989: Theorem 1) and referred to in Section 3 above. The main steps in the algorithm are as follows. (a) Start with initial values of {λ_{Bi}}, {P _{i}}, {q _{i}}, and {v _{i}}. (b) For producer i, find the value of v _{i} by maximizing the Nash product ${\Gamma}_{Bi}=\{E({U}_{i}){t}_{Bi}\}\{E({Y}^{B}){t}_{Bi}^{B}\}$, given all other contracts, {λ_{Bi}}, q _{i}, the broker's discretionary behaviour, and the individual rationality constraints (10a)–(10d). Then maximize W _{i} with respect to q _{i} given the new value of v _{i}. (Recall that λ_{Mi} = 1.) All maximization procedures utilized the grid method. (c) Repeat the process for all producers until {q _{i}} and {v _{i}} converge (to $\{{q}_{i}^{\alpha}\}$ and $\{{v}_{i}^{\alpha}\}$, say). (d) Calculate a new set of {λ_{Bi}} and {P _{i}}, using $\{{q}_{i}^{\alpha}\}$ and $\{{v}_{i}^{\alpha}\}$, and repeat steps (b), (c), and (d) until $\{{\lambda}_{Bi}\}\text{,}\{{P}_{i}\}\text{,}\{{v}_{i}^{\alpha}\}$, and $\{{q}_{i}^{\alpha}\}$ all converge to the desired equilibrium values. While no general convergence proof is provided, the algorithm actually converged when applied to the present model.
The model employed in the analysis was the one developed in this chapter. The probability distribution function of market prices was assumed to be uniform on the interval [a,b], so that P _{s}, the highest offered price in a sample of s prices observed by the broker, has the following density function: (p.311)
The handling cost functions of the broker and miller, respectively, were ${C}_{B}={\alpha}_{1}({Q}^{B}){}^{{\alpha}_{2}}$ and ${C}_{M}={\beta}_{1}({Q}^{M}){}^{{\beta}_{2}}$. The producers’ expected utility functions were those given in (23) and (24) above. The values of the environmental and structural parameters (a, b, ξ, Z, α_{1}, α_{2}, β_{1}, β_{2}, R _{i}, and Q _{i}) are given in Table 15.1, along with the assumed numbers of producers. The base case values of the parameters were selected so as to be consistent with data provided by Subbarao (1978). They do, however, involve a high degree of arbitrariness, and the results presented in the table should therefore be regarded as essentially illustrative.
Table 15.1 contains the equilibrium values of the contractual parameters (v _{i}, q _{i}, P _{i}), the broker's power coefficients (λ_{Bi}), and the parties’ expected utilities. These values are calculated for varying numbers of producers and environmental and structural parameters (‘experiments’). Unless otherwise indicated in the first column of the table, all ‘experiments’ are based on the base‐case parameters.
In the first experiment the number of producers is varied, keeping total marketed output fixed. It is evident from the table that, as the number of producers increases, the broker's power, as reflected by λ_{Bi}, increases; the brokerage fee (v _{i}) increases; and the price paid by the miller to producers declines.^{7} The expected income of the marketing intermediaries is accordingly increased. Notice that the sum of all parties’ expected utilities also rises (last column of table) with the number of producers. This is due to the increased brokerage fee, which provides an incentive for a more intense search effort by the broker.
In the second experiment, output distribution among producers is unequal. It turns out that the big producers have a stronger market position and consequently enjoy a somewhat lower brokerage fee and a slightly higher miller's price (P _{i}).
In the third experiment, there exist two groups of producers with different attitudes toward risk. It is then found that in equilibrium the low‐risk‐averse producers (R _{i} = 0.001) pay a lower brokerage fee and receive a higher miller's price. They also market a larger portion of their output through the broker. The converse is found for the high‐risk‐averse producers.
In the fourth experiment the range of price variation is broadened from (p.312)
Table 15.1 The Effect of Variation in Environmental and Structural Parameters on the Contractual Parameters, Broker's Power Coefficient, and Agents’ Utility^{*}
Environmental/structural parameter 
v _{i} 
q _{i} 
P _{i} 
λ_{i} 
E(U _{i}) 
E(Y ^{B}) 
Y ^{M} 
Σ E(U _{i}) + E(Y ^{B}) + Y ^{M} 

1. Number of producers 

4 
0.063 
221.56 
56.28 
0.890 
28 160 
1248 
2213 
116 101 
10 
0.069 
94.40 
56,43 
0.936 
11 287 
1552 
2060 
116 482 
20 
0.077 
48.97 
55.87 
0.985 
5 602 
2015 
2626 
116 681 
2. Distribution of output 

5 producers: Q _{i} = 100 
0.072 
92.91 
56.12 
0.968 
5 629 
1675 
2141 
116 571 
5 producers: Q _{i} = 300 
0.070 
96.85 
56.39 
0.959 
16 922 

3. Unequal risk aversion 

5 producers: R _{i} = 0.001 
0.071 
137.89 
56.07 
0.961 
11 247 
1732 
2647 
116 484 
5 producers: R _{i} = 0.002 
0.075 
50.14 
55.79 
0.959 
11 174 

4. Limits on the distribution of product price 

44 ≤ P ≤ 66 
0.095 
140.56 
57.57 
0.974 
11 566 
3689 
1056 
120 405 
5 Unequal risk aversion and available output 

5 producers: R _{i} = 0.001, Q _{i} = 100 
0.071 
100.00 
— 
0.963 
5 630 
2520 
1665 
116 425 
5 producers: R _{i} = 0.002, Q _{i} = 300 
0.071 
81.95 
56.00 
0.964 
16 818 
(*) Base‐case parameters: no. of producers = 10, a = 47, b = 63, ξ = 20, Z = 61, α_{1} = β_{1} = 0.01, α_{2} = β_{2} = 1.8, R _{i} = 0.01, ∀_{i}, Σ Q _{i} = 2000. Unless otherwise indicated, Q _{i} = Σ Q _{i}/n ∀_{i}.
In the fifth experiment two groups of producers are again distinguished. The first group consists of low‐risk‐averse‐small‐output producers, while the second is made up of high‐risk‐averse‐large‐output producers. It is then found that the first group markets all its output through the broker, while the second group diversifies its marketing channels, as predicted by our theory.
While the results presented in Table 15.1 by no means constitutes a test of this theory, they nevertheless suggest that it might have captured the essential features of the equilibrium set of marketing contracts.
7. Efficiency Considerations
Is the equilibrium set of contracts Pareto‐efficient? This problem in effect consists of two sub‐problems, the first related to the efficiency of the individual contract, the second to the inter‐contract relationship, that is to the effects of market organization on economic efficiency.
Let us begin with the individual contract. Notice first that, in itself, the contract between a producer and the miller is devoid of any efficiency effect, since its contractual parameters (P _{i}, Q _{i} − q _{i}) do not affect behaviour and are, thus, pure income transfer instruments. The case of the producer–broker contract is, however, different since the contractual parameters (v _{i}, q _{i}) do influence the broker's search effort. Yet, being a Nash solution, this contract is Pareto‐efficient subject to the constraints imposed by the admissable set of contracts. Accordingly, the search effort, s, is left at the broker's discretion, and brokerage fees are proportional to value of sale. However, if the cost of monitoring and enforcing s is sufficiently low, then a more efficient contract, in which s serves as a contractual parameters, could be devised. Under such circumstances, the present contract is only second‐best, as the broker's search effort is suboptimal since he ignores its contribution to the producer's expected utility. Presumably, monitoring and enforcement costs are sufficiently high to justify the elimination of the search effort as a contractual parameter in real‐world contracts. Also, given that s is not a contractual parameter, restricting brokerage fees to be proportional to sales links the risk‐sharing effect, the incentive effect, and the income distribution effect of (p.314) the contractual arrangements, thereby constraining severely the efficiency of the individual contract. Thus, contract simplicity induced by bounded rationality entails efficiency losses.
Considering, next, the inter‐contractual efficiency effects, we notice that the broker's search resembles a public good, for all producers benefit from a greater s. However, only the marginal effects of v _{i} and q _{i} on E(Y ^{B}) and E(U _{i}) alone are considered in any particular contract negotiations. The broker's search effort and the amount of output marketed through this distribution channel are therefore suboptimal. Under co‐operative marketing such externalities are, presumably, internalized, and the associated inefficiencies removed.
8. Concluding Remarks
Marketing contracts emerge under the fairly common conditions of great uncertainty and information asymmetry. (For example, only intermediaries obtain first‐hand information on buyers’ offers.) Consequently, such contracts often feature risk‐sharing and incentive provisions in addition to price‐like terms of exchange. Various specialized marketing intermediaries offer different contractual arrangements, and, given producers’ diversity, attract different producers’ groups. Thin markets along with transactors’ diversity tend to create market imperfections and idiosyncratic contracts, thus giving rise to bargaining relations, and the resulting equilibrium sets of contracts reflect the parties’ market power. When the number of transactors increases, the cores of these games shrink, and the scope for bargaining diminishes. The theory illustrated in the present analysis views each contract as a bilateral bargaining game. It characterizes the equilibrium set of contracts and allows one to explore its properties: the contractual terms, the choice of distribution channels, the division of gains among transactors, and the economic efficiency of the marketing system.
The theory is applicable to a broad spectrum of market structures, and while analytic results may be difficult to derive in more complex instances, a numerical analysis based on the algorithmic approach demonstrated in this paper should, in general, be possible.
Notes
(1.) See, e.g., Arrow (1986), Harris and Raviv (1979), and Shavell (1979).
(2.) According to Subbarao, village commission agents and millers’ agents handled 92 per cent of the marketing at the village level (Subbarao 1978; 17).
(3.) Nash solution of a two‐person (i and j) bargaining game is obtained by maximizing the product ${\Gamma}_{\mathrm{ij}}=\{E({U}^{1}){t}_{\mathrm{ij}}^{j}\}\{E({U}^{j}){t}_{\mathrm{ij}}^{j}\}$ with respect to the contractual parameters taking into account the discretionary behaviour of the players. The expressions ${t}_{\mathrm{ij}}^{i}$ and ${t}_{\mathrm{ij}}^{j}$ are the highest payoffs that the players can guarantee themselves in the event that they fail to agree. Nash solutions satisfy certain highly plausible requirements, and represent constrained Pareto optima. See Nash (1953) and Harsanyi (1977).
(4.) Both marketing intermediaries will agree to market additional output at the current contractual terms; at least up to a point. This is because, for each intermediary, the marginal change in expected income from an incremental increase in the output marketed by him is non‐negative. For if it were negative for some intermediary, the producer might offer him an acceptable reduction in the quantity marketed, which could benefit the producer. This is so even if the producer would have to dispose of this quantity at a zero price. Conditions 3 would then be violated. Hence, in equilibrium the marginal gain to each intermediary must be non‐negative, and will usually be positive. However, the additional output that the intermediary is willing to market at the given contractual terms may be limited owing to the increasing marginal handling cost. In the following it is assumed that this limit is never reached, either because the ‘measure’ of each producer is small and/or because the cost function is not too convex. Since, according to Nash bargaining theory, the threats represented by the disagreement point are never executed, an alternative interpretation is that current contractual terms determine the parties' perceptions of the relevant disagreement point.
(5.) If ${q}_{i}^{2}{\sigma}_{s}^{2}$ and ${Q}_{i}^{2}{\sigma}_{{s}^{*}\text{,}i}^{2}$ are sufficiently small, then for any U _{i}(·), (23) and (24) provide good approximations to E(U _{i}) and $E[{U}_{i}\{{Q}_{i}(1{v}_{i}){P}_{{s}^{*}\text{,}i}\}]$, respectively. The second terms on the right of (23) and (24) are then the corresponding producer's risk premia, expressed in terms of a certain income (see Arrow 1971; 96).
(6.) It can be shown that, with suitable probability distribution functions of prices, P _{i} converges to a finite value as q _{i} → Q _{i}.
(7.) The results in the second row are not exactly in line with the general trend — apparently owing to incomplete convergence of the algorithm.
Notes:
(2.) According to Subbarao, village commission agents and millers’ agents handled 92 per cent of the marketing at the village level (Subbarao 1978; 17).
(3.) Nash solution of a two‐person (i and j) bargaining game is obtained by maximizing the product ${\Gamma}_{\mathrm{ij}}=\{E({U}^{1}){t}_{\mathrm{ij}}^{j}\}\{E({U}^{j}){t}_{\mathrm{ij}}^{j}\}$ with respect to the contractual parameters taking into account the discretionary behaviour of the players. The expressions ${t}_{\mathrm{ij}}^{i}$ and ${t}_{\mathrm{ij}}^{j}$ are the highest payoffs that the players can guarantee themselves in the event that they fail to agree. Nash solutions satisfy certain highly plausible requirements, and represent constrained Pareto optima. See Nash (1953) and Harsanyi (1977).
(4.) Both marketing intermediaries will agree to market additional output at the current contractual terms; at least up to a point. This is because, for each intermediary, the marginal change in expected income from an incremental increase in the output marketed by him is non‐negative. For if it were negative for some intermediary, the producer might offer him an acceptable reduction in the quantity marketed, which could benefit the producer. This is so even if the producer would have to dispose of this quantity at a zero price. Conditions 3 would then be violated. Hence, in equilibrium the marginal gain to each intermediary must be non‐negative, and will usually be positive. However, the additional output that the intermediary is willing to market at the given contractual terms may be limited owing to the increasing marginal handling cost. In the following it is assumed that this limit is never reached, either because the ‘measure’ of each producer is small and/or because the cost function is not too convex. Since, according to Nash bargaining theory, the threats represented by the disagreement point are never executed, an alternative interpretation is that current contractual terms determine the parties' perceptions of the relevant disagreement point.
(5.) If ${q}_{i}^{2}{\sigma}_{s}^{2}$ and ${Q}_{i}^{2}{\sigma}_{{s}^{*}\text{,}i}^{2}$ are sufficiently small, then for any U _{i}(·), (23) and (24) provide good approximations to E(U _{i}) and $E[{U}_{i}\{{Q}_{i}(1{v}_{i}){P}_{{s}^{*}\text{,}i}\}]$, respectively. The second terms on the right of (23) and (24) are then the corresponding producer's risk premia, expressed in terms of a certain income (see Arrow 1971; 96).
(6.) It can be shown that, with suitable probability distribution functions of prices, P _{i} converges to a finite value as q _{i} → Q _{i}.
(7.) The results in the second row are not exactly in line with the general trend — apparently owing to incomplete convergence of the algorithm.