This paper explores the economic conditions for the viability of organic farming in a context of imperfect competition. While most research dealing with this issue has adopted an empirical approach, we propose a theoretical approach. Farmers have a choice between two technologies, the conventional one using two complementary inputs, chemicals and seeds, and the organic one only requiring organic seeds. The upstream markets are oligopolistic and the firms adopt Cournot behavior. The game is solved backward. The equilibrium distribution of the farmers between both sectors is obtained by a free-entry condition. Since multiple equilibria could exist, including the non-emergence of organic farming, we spell out viability conditions for organic farming. Then, using an “infant industry” argument, we propose several public policy instruments able to support the development of organic farming and assess their relative efficiency. Results could be useful to assess the conditions of emergence and viability of agricultural innovations in analogous contexts.
The sufficient conditions for optimality
Let us observe that the Hessian matrix of the profit function is given by with
where stands for the nth derivative. Now remember that under market clearing the amount of seeds used by an organic farmer is , the same being true for conventional farming hence . If we carry out this change of variables and introduce the elasticity of , the previous Hessian becomes
If both diagonal terms are negative, H is negative definite. Since , it remains to check that
This result is of course obvious when . So let us consider the case in which . Now let us first observe that at an optimal strategy of a Cournot player markets always clear. We can therefore say that and are the aggregated quantities of the two kinds of seeds that are supplied, so that and are market shares which belong by construction to . Moreover n and are both greater than 1 otherwise one sector would not be activated. Finally remember that we have assumed that . If we make use of the three remarks, it immediately follows that conditions  holds
Proof of Lemma 1
Let us define . It is easy to observe that:
since and . Moreover we notice that:
since and remains bounded.
Proof of Proposition 2
Let us come back to the definition of given in Lemma 1. If we now apply the implicit function theorem, we immediately observe that:
which proves the first part of the proposition. Let us now push m to infinite, the equation simply becomes since is bounded. Hence .
Proof of Lemma 3
The proof of Lemma 2 is immediate. Remember that the . So let us study . By computation we obtain
Now let us observe that:
, since we have assumed that and , and .
Proof of Proposition 4
Assume that , this means that , and we can deduce from Lemma 1 that , . If we now remember that is increasing since , we can say that , . Now remember that , this implies, in case (i), that , . It remains to mix these two observations in order to say that:
It is impossible to observe an equilibrium distribution which involves organic farming.
if and since , we can say that , . With the same arguments as in point (i) and by simply reversing the inequalities we can conclude that , i.e. organic farming always dominates conventional agriculture.
if none of these conditions is satisfied, organic farming occurs if and only if because is -shaped with respect to n.
Proof of Proposition 6
Let us recall that the outcome of our model can be reduced to three equations: the modified first-order conditions of the input providers, i.e. eqs  and the free-entry condition, i.e. eq. . These equations, after the introduction of the different policy arguments, are summarized in eq. . However to simplify the notations let us introduce , and . We can even notice that (i) see Lemma 1, (ii) , and (iii) by construction. This last point requires an additional comment. In the comparative static exercise we are looking at what happens in a neighborhood of an equilibrium which has the property that and that all policy arguments are set to 0. So by construction at the equilibrium , and since we apply the Implicit Function Theorem (IFT) from a local point of view, we can choose the neighborhoods such that at the new equilibrium.
Now let us build the function:
And since an equilibrium is given by , let us apply the IFT. By a simple exercise of computation and by bearing in mind that we observe that:
Now let us observe that the determinant of given by:
Being non-zero, we can therefore apply the IFT and we know that (at least locally). Moreover it is a matter of fact to check that:
We therefore obtain that:
Since and at an equilibrium, we can conclude that:
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