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Farm Input Supply under Contractual Insecurity
by
Matthieu Delpierre,
PhD student in Agricultural Economics, ECRU, UCL
January 2005
Keywords: interlincked transactions; local public goods; liberalisation; export crops
Abstract
In recent years, the dismantling of public schemes of export crops collection and marketing
was expected to induce efficiency and rural households’ welfare enhancement. Currently,
empirical findings tend to cast doubt upon this statement and point to harmful effects of fierce
purchase competition. In a context of contractual insecurity, the sustainability of farm credit
transactions might prove precarious. The aim of the paper is to attempt a translation of these
issues into formal terms. We focus on input supply behaviours, drawing a close analogy
between spontaneous chemicals deliveries by private traders and voluntary contributions to a
public good. Some normative stances are deduced from the analytical framework regarding
the efficient level of input or service provision and the peasants’ income.
Acknowledgements:
I am grateful to Jean-Philippe Platteau who oversaw the research which leads to the present
paper. At the writing stage, and for the provision of some mathematical subtleties, I thank
Frédéric Gaspart whose advice was also really helpful.
0
Introduction
In sub-Saharan Africa, export crops were frequently collected, treated and marketed trough
official circuits under the authority of the State. In recent years, a majority of these circuits
were dismantled and their removal was expected to induce market entries and a greater
competition at purchase and other levels of the export chain.
Parastatal agencies were responsible for providing several services to producers such as a
centralised scheme of farm credit, insurance through official prices, agricultural research and
popularisation. As a consequence of the dismantling of official agencies, the availability of
farm inputs for peasants might be dramatically reduced. More precisely, the disappearance of
the public seasonal credit scheme, aimed at financing seeds, fertilisers and pesticides, might
hurt rural households. Unfortunately, in a context of market failures and weakness of
government and judiciary regulation, particularly as far as contracts are concerned, private
intermediaries may fail to act as efficient alternative input suppliers. This unintended effect of
liberalisation is the main concern of the paper.
In recent years, agro-processing public firms came in for sharp criticism. As well as being
considered to be the place of inefficiencies and rent seeking, they are thought as inappropriate
instruments taxing producers. In addition to provide an inexhaustible income source, the close
control of export circuits allowed political leaders to build a firm foundation on which their
power could rest. Pieces of monopsonistic rent remunerated the support of the rural elite, for
example, but were also used to finance broad campaigns of co-optation in towns1.
In view of this assessment, privatisation reforms were backed by analysts and the World
Bank, among others. Akiyama and al. (2003, pp. 5-10) consider that the origin of market and
institutional reforms lie in shifts in development economists’ thought and in the resulting
change in policies.
Promotion of competition was primarily carried out for the twofold purpose of reducing
inefficiencies in collection, transformation and marketing of the crops and of increasing the
share of producer price in the world price. It seems that policy makers wished to enhance
peasants’ welfare through lowering the monopsonistic power they previously faced, a
statement which we shall, at least partially, challenge.
Shepherd and Farolfi (1999, pp. 75-76) state that reform instigators focused on the drawbacks
of preceding systems without even trying to rightly anticipate the implications of the new
ones. A basic issue which seems not having being tackled, at least initially, is the replacement
of the State in agricultural service provision. If private traders are logically expected to fulfil
the role of buyers, roles of credit, research or extension providers might not been taken on
automatically.
Numerous case studies have been conducted following such liberalisation reforms in the
developing world. Many of these studies lead to roughly similar assessments2:
First, the share of the producer price in the world price tends to grow, thereby reducing profit
margins of intermediaries.
Secondly, peasants are more directly exposed to the variability of the output world price than
in the previous system.
Thirdly, a declining quality might frequently be noticed.
1
See Boone (1992) for a well documented description of such a strategy implemented by the Senegalese
government until the eighties.
2
See Shepherd & Farolfi (1999), Poulton & al. (2004), Akiyama & al. (2003), or Gibbon (1999).
1
Finally, access to farm credit is reduced, making it difficult for a producer to purchase inputs
through the market. Input use unambiguously decreases.
According to Poulton and al. (2004), in a context of credit market failure and of asymmetry of
information about quality, the more competitive is the market structure, the more acute are the
last two problems. The authors consider that co-ordination between private traders (i.e. less
competition) might be an appropriate means of solving input use problems or of reaching
higher quality grades. Indeed, effective co-ordination between firms translates into a greater
internalisation of side-effects of individual quality control and input supply. These operations
entail positive externalities and co-ordination allows for common interest to be taken into
account to a larger extent. However, as co-ordination often entails price collusion, to the
detriment of trade efficiency, Poulton and al. introduce the idea of a trade-off between
competition and co-ordination, a conjecture highly corroborated by the facts they analyse.
They, moreover, take a stand on these alternatives as they claim3:
“Our judgement is that those sectors that achieve effective co-ordination will perform better
than those that do not. The experience of the six sectors4 to date suggests that this will be true,
even if co-ordination is achieved at the expense of some loss in competition.”
The aim of the paper is to shed some theoretical light on these facts and stances. Our approach
will allow us to form a qualified opinion on the competition / co-ordination trade-off.
The paper begins with a brief description of the salient features of the market and institutional
context in which we glide. A discussion of the sustainability of bilateral credit transactions
follows. The discussion comes to the conclusion that these contracts may be unenforceable in
the described context. Section 2 introduces our assumptions regarding peasants’ behaviour.
Our model is a first time computed in section 3 in order to highlight the effects of strategic
interaction between potential input suppliers. In section 4, some normative considerations are
drawn. The outcome is assessed regarding efficiency of equilibrium service provision, on a
one hand, and rural households’ income, on the other. Once stated the relevance and the limits
of our assumptions, section 5 concludes and brings a trail for further investigations.
Section 1 : Precariousness of Bilateral Credit Transactions
As noted earlier, we shall focus on farm input provision and credit transactions which might
support it.
To begin with, it is worthwhile to spend some time looking at the institutional environment in
which these transactions would take place. Contextual features may, indeed, affect their
sustainability as will be briefly reminded. Subsequently, we examine strategies which might
support interlocking transactions of input supply, credit, and output purchase.
These concerns affect our issue directly. Indeed, if private intermediaries are required to fill
the economic space left by the withdrawal of the State, the security of contractual
relationships becomes a matter of the utmost importance. In a liberalised sector, the
institutional context, which might affect the sustainability of credit transactions, notably, is a
question to address.
Regarding economic transactions, the context comprises two relevant dimensions: the market
and the regulatory environments.
3
4
Poulton & al. (2004, p. 523)
the cotton sectors of Ghana, Mozambique, Tanzania, Uganda, Zambia and Zimbabwe
2
Regarding the first one, the extent of transaction costs in the African countryside seriously
affects market exchanges. Filling the informational gap is often prohibitively costly for a
potential creditor. The atrophy of credit markets straight follows.
Secondly, as far as public regulation is concerned, the administrative and coercive capacities
of the State are quite weak in general. Furthermore, to obtain the judicial settlement of a
contractual dispute, the plaintiff incurs time and money, costs which will probably exceed the
gain of resolution. One could argue that a physical collateral might be set up to protect the
creditor of a loan, for example. But, on the one hand, property rights are sometimes
insufficiently defined to make this arrangement possible, and, on the other hand, households
are often not wealthy enough (or do not possess valuable livestock, large land endowment or
farm machinery) to establish any collateral.
Once these contextual elements are taken into account, their implications can easily be
highlighted.
First, the farmer faces a binding liquidity constraint resulting from credit market
imperfections. As a direct consequence, his input purchases are potentially restricted.
Second, the judicial option is not credibly available because having recourse to it is either
extremely costly (in relation to money at stake) or not possible at all. It follows that both
contracting parties should face the right incentives, those which will induce them to fulfil their
reciprocal obligations. This being said, it leads us to define the contractual insecurity. We will
refer to an unsecured transaction as a contract whose clauses are not binding per se, except for
the psychological cost of reneging on the agreement. The strategic rationality is therefore
logically assumed in a game theory framework throughout the paper. Promises or threats will
only be taken into account if they credibly can be implemented.
Turning to input credit transactions, we briefly examine their sustainability under such a
context. Interlocking transactions of input supply, credit and output purchase occur in several
parts of the world5. It reflects that they may prove profitable for those who offer such
contracts. A twofold argument lies behind this profitability6.
On the one hand, private traders often own specific assets downstream from growing, such as
warehouse facilities or information networks, an incentive to offer input credit which can
contribute to secure these investments by increasing the volume harvested and purchased.
On the other hand, compared with any other potential credit supplier in the countryside, the
usual buyer of the crop incurs lower levels of transaction costs. He enjoys a better knowledge
of his debtor, while monitoring is also made easier by the loan being granted in kind. And,
finally, the recovering takes the form of a deduction from the market value of output, easily
observed by the usual trader.
Interlocking transactions schemes involving credit are sustained in practice, but not
everywhere. It is therefore useful to highlight the conditions under which the creditor can or
can not protect himself against opportunistic behaviours. In particular, the trader has to
prevent his debtor from side-selling. The farmer may, indeed, selling his output to a
competitor, fail to repay his debt.
It emerges from observations that interlocking transactions take place in long term
relationships. It is needed to distinguish between two cases: monopoly provision of input
credit and multiplicity of suppliers.
5
See Smith, Stockbridge & Lohano (1999) in the case of Pakistan, and Dorward & al. (1998) for two other
African cases.
6
These arguments appear notably in Dorward & al (1998).
3
If only one trader-creditor comes to the village, the threat of withdrawal of credit facilities can
obviously lead peasants to repay their debts, the shift of creditor being an unavailable option 7.
If, on the contrary, there exist several credit sources, the recovery of the loan is more
problematic. According to repeated game theory, nonetheless, a repayment outcome may be
reached if strategies involving reputation mechanisms as defined by Greif (1983) are
implemented.
On the one hand, if it is costly for the peasant to build confidence with a trader, the
breakdown of the acquaintance following a deviation can be an incentive to behave honestly.
It seems to be the case in Pakistan8, for example. Unfortunately, in sub-Saharan Africa,
industrial over-capacities foster strong competition among traders and prevent them from
inflicting any cost of entry to their suppliers. The bilateral reputation mechanism does not
apply.
On the other hand, if a black list of bad debtors circulates among intermediaries (multilateral
reputation mechanism), it enables them to activate a collective punishment, whereby
defaulting borrowers are excluded from the whole market. But nothing ensures that this
equilibrium will actually occur as firms may easily fall in a prisoner’s dilemma9, not share
information and compete in prices. The model will fit into this scheme.
To sum up, let say that bilateral credit transactions are difficult to sustain under fierce
competition among traders. In other words, the probability of emergence and the viability of a
private, decentralised input credit scheme is a decreasing function of the level of competition.
In what follows, in the model and its developments, we assume factors market failures and
unsecured transactions, according to a realistic assessment of the institutional context. We
also take explicitly into account the complete failure of the rural credit market. In order to
acquire seeds and fertilizers, peasants exclusively rely on the good will of intermediaries.
Finally, we suppose that conditions of interlinked credit and marketing contracts’
sustainability do not hold, mainly because of a lack of co-ordination between traders. The
reputation equilibrium has not been selected, at least in the short run.
Section 2: Behavioural Assumptions
In order to put in place our model, we begin with the main behavioural assumptions. The first
is related to input allocation choices, the second to output supply behaviours.
Under unenforceable input credit contracts, traders can not prevent their debtors from sideselling. Not being able to enforce the exclusive right to purchase they theoretically enjoy,
input suppliers compete against each others as equals for crop acquisition.
Furthermore, they can not monitor the allocation of seeds or chemicals they provide, hence
having no means to influence input application. Peasants may therefore divert fertiliser from
their own plots. More precisely, the origin of diversion will be the disparity of initial input
endowments. Initially better endowed producers, those who receive more chemicals from
traders for a reason or another, will redistribute part of it to fellow villagers until the marginal
productivity of fertiliser will be equalised over all plots. Before this equalisation is achieved,
there remain mutually profitable agreements to conclude. We have seen earlier that the
7
For appropriate values of the discount factor (future income being sufficiently weighted by the producer)
Smith, Stockbridge & Lohano (1999)
9
In transition sectors, and typically following liberalisation reforms, the market structure is unstable, entrants do
not know each others and a lack of co-ordination is highly plausible.
8
4
liquidity constraint was particularly binding. One could therefore raise the objection that those
who benefit from this redistribution may have difficulties to provide a monetary counterpart.
Yet, given that they often live near to each other, it is quite easy to find other kinds of
remuneration such as services in other spheres of social life. It is well known that the rules of
the community have the ability to protect these transactions. We will refer to this behaviour
speaking about intra-village arbitrage. The translation of this assumption in mathematical
terms is quite easy. The quantity harvested over all plots of the village is an increasing
function of the total quantity of inputs provided by the traders ( e ) with a declining marginal
impact, amounts of land and labour being given as will be explained below:
Q (e) with Q' (e)  0, Q(e)  0
This specification is grounded on an homogeneous application of fertiliser. To be convinced
of it, you can notice that it, on the contrary, does not represent an heterogeneous one. Indeed,
if each producer had received the same amount and applied it to his own plot, villagers being
supplied one after another, we would have written:


Q(e)   Qi (ei ) with Qi (ei )  0, Qi (ei )  0, i ,
i
the index being related to each individual producer,
but the facts square only with the former specification where total input affects the whole
village production regardless of its initial apportionment.
Let us now turn to output supply behaviours. The producers do not anticipate the output price
when allocating their land and labour force. The total quantity harvested in the village is
therefore only determined by the amount of fertiliser delivered, provided that production is
assumed non carrying any climatic risk. At the time of marketing the crop, the quantity
harvested is given. Supply behaviours are grounded on marketing costs villagers face instead
of being generated by production costs. This allows to avoid concerning about the values of
medium and long term price elasticities of agricultural production which are controversial. In
practical terms, peasants have the option of going themselves to local markets, incurring
transport costs, instead of channelling their output through intermediaries. The marginal cost
of self-marketing is assumed to increase linearly at the village level. We translate it formally
as follows: suppose a transport cost uniformly distributed in the interval t L , t H  10. The
prevailing market price of output is exogenously given, as a consequence, we define the range
of reservation prices when sales decisions are taken in the village: p̂ is uniformly distributed
in the interval  pˆ L , pˆ H  , with
pˆ L  pu  t H
pˆ H  pu  t L
where pu is the prevailing market price. This expositional device makes interpretation easier
as you can see an individual producer behind each of these costs and associated reservation
prices. According to this interpretation, peasants would be homogeneous in all respects
excepting their self-marketing cost.
The inverse supply function is written:
p(q)  pˆ L  b.q(e)
(1)
where
b0,
10
The transport cost is not stochastic. The uniform distribution is a distribution of types among the peasants
population. These costs are their self-marketing costs, to be incurred if they decided not to channel their crop
through intermediaries.
5
is defined on the support 0, Q(e) and is perfectly inelastic in Q (e) which is the harvest size
over the whole village.
pˆ  pˆ L
Q (e)  H
b
The supply function describes the peasants’ decision of either to sell the output to traders in
the village or to market it themselves. Given their heterogeneity and a purchase price between
the two bonds  pˆ L , pˆ H  , a fraction of total output will be channelled through traders, the rest
will be sold on the domestic market by the villagers themselves.
These informations are summed up graphically on figure 1.
Figure 1. The Supply Curve at the Village Level
p
p̂ H
p̂ L
0
Q (e)
q
Peasants, or more precisely their crop, are placed in descending order of self-marketing cost
on the horizontal axis.
Having defined the behaviour of the villagers regarding the allocation of farm inputs and sales
decisions, we turn to the model.
Section 3 : The Duopsony Case
Before starting the game, let us define the players, the strategy spaces and the chronological
structure.
Two export crop traders compete with each other. Their access to capital enables them to
acquire and distribute fertiliser at the start of the season. Before harvest time, they choose
their purchase, transport and treatment capacity which will remain immutable at marketing
time.
6
Their strategy space therefore comprises three dimensions, chronologically listed:
ei  0, ,
the amount of fertiliser delivered, measured in money11,
qi  0, ,
the transport capacity, acquired at a constant marginal cost,
pi  0, ,
the unit price they charge for output purchase.
In addition, we suppose that the constant marginal receipt of intermediaries at the village level
is exactly the same as the one of the most efficient self-marketer among villagers ( p̂H ). This
means that both traders face the same marginal transport cost as this peasant. This assumption
is useful in order to heavily simplify the mathematical treatment of the problem without
modifying the main conclusions of the analysis.
It has been shown that a Bertrand competition under capacity constraint amounts to a Cournot
competition12. This result is also known as the Edgeworth solution. As a consequence, the two
last stages of the game reduce to a single stage in which both players choose the quantity
purchased, given the supply curve at the village level. Would we have straight assumed a
Cournot competition, which has the appeal of making the results nice and meaningful, the
outcome would have been the same. But it seemed useful to justify this choice. The price is
indeed, as a last resort, in the hands of the purchaser, while quantity constraints and weak
ability of capacity adjustment are highly plausible at the marketing stage.
The rigid capacity constraint assumption might appear to be rather strong, but it could be
relaxed to some reasonable extent without altering the outcome, as will be mentioned below.
The game will be solved using backward induction. Once equilibrium quantities (q1 , q2 ) will
have been found for each pair (e1 , e2 ) at the close of the first stage of the game, this
information will be included into the payoffs of the fertiliser game.
Equilibrium Quantities, the Second Stage
Making use of (1), the optimisation problem of trader 1, for given (e1 , e2 ) and q 2 , can be
written as:
MAX 1   pˆ H  p(q1  q2 ) .q1  e1   pˆ H  pˆ L  b.q1  q2 .q1  e1
q1
The objective is defined as the profit margin made on quantities purchased minus the total
transport cost and costs of capacity acquisition, already deducted when writing p̂H , and the
fertiliser “investment”.
When maximising for q1 , we find the reaction function of trader 1:
pˆ  pˆ L  bq2
q1* (q2 )  H
2b
and similarly for trader 2.
11
This sum represents a quantity because inputs are valued at the prevailing market price of fertiliser. No fee is
explicitly charged for input provision given that peasants do not possess any liquidity and that credit transactions
are supposed to be unenforceable.
12
We refer to Kreps & Scheinkman (1983).
7
The Nash equilibrium lies at the intersection of reaction functions:
1 pˆ H  pˆ L 1
q 
 Q ( e)
i 3
b
3
2
q1  q 2  q D  Q(e)
3
1
2
 p D  pˆ L  pˆ H
3
3
the price being found through the supply function (1).
The share of total output channelled through intermediaries is two third. The equilibrium price
is a weighted sum of the two extreme reservation prices. Facing this price, peasants
characterised by high self-marketing costs, those “located” on the left of the supply curve,
will sell their crop to one of the traders, the others will transport and market their crop
themselves.
The Fertiliser Game, the First Stage
Including equilibrium quantities or prices, the optimisation program of each player can be
written and solved:
1
 pˆ H  pˆ L  1
D
. Q(e1  e2 )  e1
MAX 1   pˆ H  p (q ) . Q (e1  e2 )  e1  
3
3

3
e
1
 1
9
 0  Q' (e1  e2 ) 
e1
pˆ H  pˆ L
from which the reaction function is deduced to have the form:
e1* (e2 )  e *  e2
where


9

e  Q1 
 pˆ H  pˆ L 
similarly for player 2. Figure 2 illustrates this result graphically in the input contributions
space.
Figure 2. Reaction Curves of Input Delivery
e1
e*
e1* (e2 )  e *  e2
e2* (e1 )  e *  e1
0
 9 
e  Q' 

 pˆ H  pˆ L 
*
e2
1
8
The two reaction curves are superimposed. There exists an infinity of Nash equilibria located
on the locus of couples (e1 , e2 ) the sum of which is a constant equal to e * . Each of these
equilibria defines a sharing out of fertiliser purchases and deliveries among traders. Nothing
ensures that they will contribute equally.
Nevertheless, the infinity of equilibria share a common feature: a same amount of input
delivered and applied to the crop13. This amount, optimal from the intermediaries’ point of
view, looks like a public good, for the three following reasons.
First, given that players compete against each others as equals for crop acquisition, the total
amount of fertiliser delivered benefits them both through a larger harvest. Input investment
actually generates positive externalities from a buyer to the other, and to the peasants, as
shown below. This outcome is obviously closely related to the intra-village arbitrage
behavioural assumption as will be briefly discussed later.
Secondly, a close analogy can be drawn between the fertiliser game and a problem of
voluntary contribution to an indivisible public good. When studying villagers’ labour
provision in a drain project in Ethiopia, Gaspart & al. (1998) build a theoretical framework
whose the outcome is quite similar to the one of the first step of our analysis. In their model,
indivisibility emerges from the fact that a certain amount of labour provision is required for
the good to be produced. As a consequence, reaction functions of villagers exhibit some
identical properties to those of our buyers: each of them accepts, to some extent, to fill the gap
between the others’ total contribution and the required one, and strictly no more. In the case
of input delivery, the provision required is the optimal amount of fertiliser from traders’ point
of view. The indivisibility is therefore imaginary. Nevertheless, the comparison goes further
as the stake of the fertiliser game is a distribution of costs. Facing multiple Nash equilibria,
game theory does not bring any relevant selection procedure. In the Ethiopian “drain game”,
the authors conducted an empirical study with the aim of highlighting the determining factors
of labour distribution among beneficiaries. Their econometric investigation reaches the
conclusion of a statistically significant link, obviously positive, between individual labour
provision and the private benefit derived from the public good, leading to a presumed focal
point in labour distribution. In the context of our model, preferences, or, more appropriately,
cost structures of intermediaries, are assumed to be strictly homogeneous. Therefore, the
empirical finding of a link between contributions and private profitability does not apply, in
the absence of benefits variability, leaving us with an insoluble uncertainty about cost
distribution. However, this is not disturbing, the point being elsewhere. The total amount of
input provision is known, which is the only information needed to proceed further with the
analysis. How the equilibrium selection is performed at this stage is no concern of ours.


9

e  Q1 
ˆ
ˆ
p

p
L 
 H
Thirdly and finally, this sum of spontaneous contributions proves to be too low compared
with the social optimal level of input investment. These types of considerations are tackled in
the following section but usefully serve here the comparison between farm input supply under
contractual insecurity and voluntary contributions to a public good.
13
These conclusions remain true under the following condition:



9
9
 pˆ  pˆ L  1  1 
  Q  1 
  0 ,
 (e * )  0   H
. Q Q  
i
ˆ
ˆ
ˆ
ˆ
3
p

p
p

p

3 
L 
L 
 H
 H
meaning that one of them bearing the all burden of input deliveries belongs to the set of Nash equilibria.
Otherwise, if this condition does not hold, the range of Nash equilibria is reduced. Nevertheless, whatever the
assumption made, it remains that the sum of voluntary contributions is known, unless fertiliser application is not
profitable at all for traders. We suppose it is.
9
Section 4 : Optimal Market Structure
In this section, our framework is applied to the general case of an exogenous number ( n ) of
players. The number of competitors is held constant in the analysis and treated as exogenous.
You may, however, consider this number as the zero profit one. Indeed, trade operations in
the African countryside entail fixed costs such as vehicle acquisition, information collection
and so on. The regulator has the ability to influence this number through altering these costs
via licences or subsidies in order to near the optimal market structure to be derived from the
framework.
This generalisation allows to assess the efficiency of the fertiliser game outcome, on the one
hand, and to raise some welfare considerations regarding the peasants’ agricultural income, on
the other hand.
Let first express the equilibrium price, quantity, and fertiliser provision as functions of the
number of intermediaries.
Equilibrium Quantities, the Second Stage
The reaction function has the form
pˆ H  pˆ L  b.  q j
j i
q* (  q j ) 
i j i
2b
Provided that traders are identical regarding their cost structure, the outcome will be
symmetric, hence
pˆ H  pˆ L  b( n  1)q
i
q 
i
2b
1 pˆ H  pˆ L
1
q 

Q (e)
i n 1
b
n 1
n
 q O ( n) 
Q (e)
(2)
n 1
1
(3)
 pˆ L  npˆ H 
 p O ( n) 
n 1
where q O and p O denote the oligopsony equilibrium quantity purchased and price,
respectively. The share of total harvest channelled through intermediaries increases with their
number, correlatively with the prevailing purchase price at the village level. This is far to be
surprising since the level of competition rises among traders. Under perfect competition, they
buy at their marginal receipt and the whole harvest is marketed through their good offices:
n
O
Q (e)  Q (e)
lim q (n)  lim
n  
n   n  1
1
O
 pˆ L  npˆ H   pˆ H
lim p (n)  lim
n  
n   n  1
10
The Fertiliser Game, the First Stage
Using equilibrium quantities (2), each player maximises for e :
n
 1

Q( e j )  e
. pˆ H  pˆ L .
MAX i   pˆ H  pˆ L 
i
j
n 1
 n 1

e
i
1
Q( e )  e
 MAX    pˆ H  pˆ L .
i
n  12 j j i
e
i
2

i  0  Q' (e)  n  1
(4)
pˆ H  pˆ L
e
i
hence,
 n  12 

e* (n)  Q'1 
(5)

ˆ
ˆ
p

p
H
L


which represents the spontaneous fertiliser provision and is a decreasing function of n 14, the
more numerous the buyers, the less the input delivered.
Two effects can be distinguished as the number of players rises: on the one hand, as price
goes up, the profit margin over all buyers is reduced. Therefore, the amount of fertiliser
maximising the aggregate profit margin goes down. On the other hand, the aggregate margin
being given, the individual profit decreases. Indeed, when choosing their level of input
contribution, intermediaries only take into account a fraction of the aggregate profit in their
decision15. This is the effect of strategic interaction.
Having expressed the quantity purchased (2), the price (3), and the fertiliser delivered (5) as
functions of the number of traders, we now turn to some normative considerations.
The spontaneous total amount of input delivered is first discussed. Trying to assess the social
efficiency of the chemicals provision, we have to choose a benchmark. To this end, we define
a kind of social surplus as the sum of the peasants’ agricultural income and the aggregate
profit margin of traders16. The former comprises two components: the income from sales to
traders in the village and the one from the self-marketed crop net of transport costs. From the
latter transport costs are the same way deducted as well as the total value of input purchases
and deliveries. Figure 3 illustrates this “surplus disintegration”. Notice that oligopsony entails
some efficiency losses at the marketing level. Indeed, the most efficient self-marketers among
producers, those located on the right of the supply curve, incur a higher average transport cost
than intermediaries. This excess of costs is the marketing dead weight.
14
Indeed, since
n  12
pˆ H  pˆ L
increases with
n and given that Q(e)  0
Furthermore, they do not take into account the villagers’ income.
This is not strictly a surplus as far as the entire agricultural receipt is counted. To be more rigorous, we should
have deducted the remuneration of factors such as labour, land, and small farm equipment. Nevertheless, we did
not, to simplify expressions and given that factors use is held constant by assumption. See section 2.
15
16
11
Figure 3. Surplus Disintegration
p
Peasants’ Agricultural Income
Aggregate Profit Margin of Traders
Dead Weight
p̂ H
1
 pˆ L  npˆ H 
n 1
p̂ L
0
n
.Q (e)
n 1
Q (e)
q
Making use of figure 3, we work out the surface area of the dead weight:
1
n
1

 pˆ L  npˆ H 
DW (n)  . Q(e) 
Q(e) . pˆ H 
2
n 1
n 1


 pˆ  pˆ L 
1
(6)
 DW (n)  Q(e) H
2
n  12
Our social surplus is therefore defined as the surface area of the rectangle with breadth of
Q (e) and height of p̂ H minus the marketing dead weight (6) and the fertiliser cost:
 pˆ  pˆ L   e
1
SS (n, e)  pˆ H .Qe   Qe  H
(7)
2
n  12
Once it’s derivative with respect to e is obtained, we find the optimal amount of input
provision:
 pˆ H  pˆ L   e
1
MAX SS (n, e)  pˆ H .Q(e)  Q(e)
2
n  12
e
SS (n, e)
2n  1
 0  Q ' (e) 
2
e
2n  1  4n pˆ H  pˆ L
You can easily check, by comparison with the fertiliser game outcome (5), that
2
n  12 
2n  1
pˆ H  pˆ L 2n 2  1  4n  pˆ H  pˆ L
2


12
 e (n)  arg max SS (n, e)
(8)
e
provided that the production function exhibits decreasing returns. In addition, the left hand
side of (8) increases with n , while the right hand side decreases as n rises17. From these two
expressions, we deduce that (a) the spontaneous level of input provision is sub-optimal and
that (b) sub-optimality gets worse with a great number of traders. These are other ways of
making the comparison with the voluntary contribution to a public good.
Let us briefly return to the marketing stage and point that the marketing dead weight is
reduced as the exogenous number of buyers increases and is completely removed under
perfect competition. However, keep in mind the above result that input provision is lower
under fierce competition, as well as quantity harvested, suggesting a trade-off in the number
of players regarding efficiency concerns. At this stage, if we wrote, using our framework, the
optimisation program, we would reach, to some extent, the formal representation of the coordination / competition trade-off highlighted by Poulton & al. (2004) and quoted above. It
seems they focus on efficiency considerations. This question of the socially optimal market
structure is left aside though. Indeed, the reasoning is closely similar to the one developed
below. We arbitrarily chose to focus on peasants’ living standards. This choice is also
motivated by the wish to challenge political stances, as it is maintained by some that
liberalisation reforms are designed to enhance rural households’ welfare.
Let us now turn to this question. Which market structure maximises farmers’ income?
Making use of figure 3, we work out the peasants’ income:
 pˆ L  pˆ H  n  2  pˆ H  pˆ L 

PI (n)  Q(e (n)).


 .
(9)
2
2
 n  1 


This expression is easily interpretable. This income is the product of a quantity and a price.
The quantity is the size of the harvest. The price has for basis the average reservation price
and is increased by the presence of intermediaries in the village at harvest time.
In order to derive the optimal market structure from the villagers’ point of view, we take the
derivative of (9) with respect to n :
2
n pˆ H  pˆ L 
PI (n)
e  pˆ  pˆ H  n   pˆ H  pˆ L 
,
(10)
 Q' (e). . L

  Q(e).
 .
n
n 
2
2
n  13
 n  1 

where18
n  1 .
e
2

.
n Q' ' (e)  pˆ H  pˆ L 
You can easily check that the first term on the right hand side of (10) is negative 19. It shows
that, for a given average price, the peasants’ income decreases as the number of competitors
rises. The amount of input provided being reduced, the quantity harvested goes down. The
17
The benchmark varies because the marketing dead weight is a decreasing function of
ideally granted is therefore higher, given that the social surplus it maximises is greater.
18
Indeed, making use of condition (4) :
19
Given that Q(e)  0 ,
Q(e)  f (n)  Q(e).de  f (n).dn 
n . The input provision
de f (n)

dn Q(e)
e
 0 and the rest is positive.
n
13
second term is positive, meaning that, for a given harvest size, a higher number of buyers
fosters competition leading to purchase price increases, to the benefit of the villagers.
To conclude the technical analysis, let us briefly recall the main results we should keep in
mind before introducing possible further exploitations of the framework.
Section 3 drew an analogy between farm input provision and voluntary contribution to a
public good. Both entail externalities and result in a cost sharing. In this section, we saw the
inability of free input supply behaviours to reach an efficient level of total deliveries, all the
more as competitors are numerous. But once confronted with the beneficial price effect of
competition, this face of sub-optimality fades a bit regarding the villagers’ welfare. The
optimal market structure from their point of view lies somewhere between perfect competition
and monopsony.
Unfortunately, the equilibrium point in the trade-off might be rather sensitive to contextual
and technical features whose estimation might require empirical studies. The point is that
there is, a priori, no unique and universal solution, at least theoretically. It follows that stances
such as competition has to be promoted, or conversely restricted, at all costs, may potentially
be revised.
Section 5 : Generalisation, Extensions, and Concluding Remarks
The robustness of the model is the first topic we tackle in this conclusion. Some of our
assumptions may appear rather strong. Are they essential steps to reach the outcome we have
presented? Let us examine three of them.
First, we have recourse to a Cournot competition among traders while, in practice, the
operative choice of buyers is unquestionably the purchase price. Kreps and Scheinkman
(1983) have shown that capacity constraint leads price competition to the Cournot outcome.
But, according to their analysis, we could unleash our strict capacity constraint, replacing it by
an increasing marginal transport cost assumption, without affecting the outcome of a
progressive decrease in input provision as the number of players rises. Suppose that the trader
has to acquire a vehicle before harvest time, and that his transport capacity might only be
adjustable at an increasing marginal cost, the cost of additional labour employed in handling
to do one round trip more a day, for example.
Second, is it reasonable to assume that peasants have different abilities to transport and
market their crop themselves? Could the most efficient ones not sell their services to fellow
villagers? We dodge this objection reminding that the self-marketing cost heterogeneity is
designed as an expositional device intended to make interpretations easier. We could
costlessly replace it by an increasing marginal self-marketing cost assumption over the whole
village, which is highly plausible.
Third, let us mention the intra-village arbitrage hypothesis which, we confess, is a bit less
innocent. The redistribution of inputs among villagers entails an uniform application over all
plots whoever chemicals come from. This assumption is obviously a mechanical source of
externalities. Yet, the relevance of our framework does not depend on the occurrence of these
behaviours, at least not critically. Indeed, the assumption translates one consequence of
contractual insecurity among a set of plausible ways of breaking the agreement. Our
framework applies to a situation in which one of the two following assumptions holds: either
intra-village arbitrage, or purchase competition among traders. Consider the former situation.
We mean redistribution behaviours occur, while farmers abide by the exclusive right to
purchase of the input provider. Externalities obviously remain as the contract is broken
upstream. In the latter situation, input contractual allocation rules are followed, but, at the
14
marketing stage, competition prevents traders from storing up the gains from their investment.
They are confronted, as in our model, with a fixed total size of the harvest at the whole village
level without being able to assert their right to purchase higher quantities because of a greater
contribution to output.
In the last resort, the public good characteristic of input provision arises from the fact that the
supplier might not store up the whole return from his investment whether it follows a sideselling or another kind of deviation such as input diversion. An environment of high level of
transaction costs, weakness of government and judiciary regulation, and fierce competition
among players will very probably prove to be a context in which bilateral credit transactions
can not be honestly conducted at equilibrium. These assertions lead us to collect our thoughts
towards a conclusion of the paper.
The instigators of liberalisation reforms focused on the efficiency enhancement in
intermediary stages of collection and transformation and on purchase price increases
competition was expected to induce. However, the dismantling of parastatal agencies and
especially the public scheme of input delivery they previously performed brings about new
issues, notably as far as service provision is concerned. Our framework shows that private
intermediaries may fail to act as alternative credit and input suppliers, or, if they do not, their
level of contribution is inefficient, according to the public good characteristic of input
provision. Notice, however, that, even if we assert the problem in terms of market structure,
the ultimate source of under-investment in seeds and chemicals lies elsewhere. Indeed,
failings in rural credit markets are responsible for externalities from traders to farmers.
Suppose villagers can afford input purchases, they would be those who make the decision of
the amount of input applied to the crop. This situation would be more efficient, at least under
perfect competition, having in mind that externalities would work in the opposite direction,
from the decision of the peasants to the profit margin of intermediaries, unless the latter is nil.
Let us now turn to few concluding remarks.
First, notice that other services exhibit public good characteristics, such as quality control,
research and extension, or popularisation. Our model therefore under-estimates the extent of
the harmful impact of fierce competition on efficiency and peasants’ welfare.
Second, Poulton and al. had in mind these effects when making a stand on the optimal market
structure. In order to weaken sub-optimality in service provision, they recommend a high
level of co-ordination, which means an extreme position on the collusion / competition
spectrum. Section 4, in which we derive the optimal market structure from the producers’
point of view, suggests to be careful with this kind of assessment. Indeed, contextual as well
as technical features may affect the location of the optimum.
Third and finally, a strict withdrawal of the State is questionable. Without casting doubt upon
the inability of governments to perform themselves all the economic operations around export
crop production and marketing, it is reasonable to think about some tasks of which the public
sector could remain in charge, the more obvious being agricultural research and
popularisation of its results. Other tasks such as money lending or direct input provision may
turn into instruments of privilege allocation and prove more sensitive. It would be interesting
to wonder about the role of farmer associations in credit and input deliveries. Indeed, these
groups benefit from several advantages such as (almost) perfect information and the ability to
set up strategies including effective punishment, any form of social sanction up to
ostracisation. In practice, following liberalisation reforms, some peasants organisations
already get involved in operations of collection, transport and input delivery on credit, in
Senegal, for example. It would probably be worth to look into such initiatives.
15
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17