Can cooperatives influence farmer’s decision to adopt organic farming? Agri-decision making under price volatility

Abstract

With the growing importance of sustainable farming and increasing fluctuations in the price of agricultural produce, the choice of nature of farming and participation in a cooperative has become critical. This paper examines farmers’ decision of adopting organic farming and participating in cooperative institutions to market their produce. We formulate a two-stage strategic game model whereby two farmers first choose a technique of production of their crops followed by a decision regarding the mechanism by which to sell their products to cope with the environment of uncertain agricultural prices. We extend the two-stage process to find out conditions under which it would be profitable for a farmer to produce organic crop. We found that farmers are more likely to produce organic crop if they can sell their produce through a cooperative. Our analytical results show that incremental costs of organic production, the operational cost of running cooperatives and crop’s price volatility can be crucial in influencing farmers’ choice of production techniques of and marketing institutions. In particular, we found that when it is easier for farmers to participate in cooperative, they tend to choose organic production technique. To empirically support the findings, we analyzed the weekly transactions of 65 Fruits and Vegetables during 2017 in six different regions in the United States. We found that regions with higher number of cooperatives registered higher transactions in organic crop.

Appendices

Appendices

1.1 Appendix 1

The prices are assumed to be normally distributed. Therefore, profit received by farmers also follows normal distribution with mean $\varPi _\mu$ and variance $\varPi _\sigma ^2$.

We define the expected utility as follows:

$$\begin{aligned} \begin{aligned} E(U)= \int f(\varPi ) d\varPi \end{aligned} \end{aligned}$$

(6)

Where $f(\varPi )= \left[ \frac{1}{(2\pi \varPi _\sigma ^2)^{1/2}}\right] e^{-\left( \frac{(\varPi -\varPi _\mu )^2}{2\varPi _\sigma ^2}\right) }$ represents the PDF (probability density function).

Hence,

$$\begin{aligned} E(U)&= \frac{1}{(2\pi \varPi _\sigma ^2)^{1/2}} \int e^{-\left( \frac{(\varPi -\varPi _\mu )^2}{2\varPi _\sigma ^2}\right) } (-e^{\alpha \varPi }) d\varPi \nonumber \\&=-\frac{1}{(2\pi \varPi _\sigma ^2)^{1/2}} \int e^{-\left( \frac{(-\varPi ^2-\varPi _\mu ^2+2\varPi \varPi _\mu -2\alpha \varPi \varPi \sigma ^2}{2\varPi _\sigma ^2}\right) } d\varPi \nonumber \\&= -\frac{1}{(2\pi \varPi _\sigma ^2)^{1/2}} \int e^{-\left( \frac{-(\varPi +(\alpha \varPi _\sigma ^2 - \varPi _\mu ))^2 + \alpha ^2\varPi _\sigma ^4 - 2\alpha \varPi _\sigma ^2\mu }{2\varPi _\sigma ^2}\right) } d\varPi \nonumber \\&= -e^{\frac{\alpha ^2 \varPi _\sigma ^2}{2-\alpha \varPi _\mu }}\int \frac{1}{(2\pi \varPi _\sigma ^2)^{1/2}} e^{-\left( \frac{-(\varPi +(\alpha \varPi _\sigma ^2 - \varPi _\mu ))^2}{2\varPi _\sigma ^2}\right) } d\varPi \nonumber \\&= -e^{\frac{\alpha ^2 \varPi _\sigma ^2}{2-\alpha \varPi _\mu }} \end{aligned}$$

(7)

We take the monotonic transformation of Eq. (7) as follows:

$$\begin{aligned} E(U)&= \varPi _\mu - \frac{\alpha \varPi _\sigma ^2}{2} \end{aligned}$$

(8)

$$\begin{aligned} \varPi _\mu =&E(P_\gamma )q_{\gamma \delta } - a q_{\gamma \delta }^2 - C_\gamma (q_{\gamma \delta })- C_\delta \nonumber \\&= P_\gamma ^\mu q_{\gamma \delta } - a q_{\gamma \delta }^2 - C_\gamma (q_{\gamma \delta })- C_\delta \nonumber \\ \varPi _\sigma&= E(\varPi ^2) - [E(\varPi )]^2 \nonumber \\&= E( P\gamma q_{\gamma \delta } - a q_{\gamma \delta }^2 - C_\gamma (q_{\gamma \delta })- C_\delta )^2 - \left( E(P_\gamma )q_{\gamma \delta } - a q_{\gamma \delta }^2 - C_\gamma (q_{\gamma \delta })- C_\delta \right) ^2 \nonumber \\&= [E(P\gamma ^2) - (E(P\gamma ))^2]q_{\gamma \delta }^2 \nonumber \\&= \sigma ^2 q_{\gamma \delta }^2 \nonumber \\ E(U)&= P_\gamma ^\mu q_{\gamma \delta } - a q_{\gamma \delta }^2 - \frac{\alpha }{2} \sigma _\gamma ^2 q_{\gamma \delta }^2 - C_\gamma (q_{\gamma \delta })- C_\delta \end{aligned}$$

(9)

Equation (8) denotes the expected utility of each farmer.

1.2 Appendix 2

1.2.1 Both grow non-organic crop

1.2.2 Both farmers sell to a cooperative

We define the expected utility of farmer selling through cooperative as follows:

$$\begin{aligned} \begin{aligned}&\underset{q_{NC}}{\text {maximize}}&E(U(q_{NC}; P_\gamma , \alpha , \sigma _N)) = {P_N^\mu } q_{NC} - a{q_{NC}}^2 - \frac{\alpha }{2} \sigma {_N}^2 {q_{NC}}^2 - \frac{F}{2} \end{aligned} \end{aligned}$$

(10)

where ${P_N^\mu }$ denotes the average price of non-organically produced crop, $\sigma {_N}^2$ denotes the variance of the price of non-organically crop, $q_{NC}$ is the amount of crop sold by each farmer through cooperative, and F represents the operational cost of the cooperative. Both farmers are assumed to be homogeneous, therefore, each farmer shares half of the operating the cooperative.

By substituting the value of $\sigma {_N}^2 = 0$ in Eq. (10), we get:

$$\begin{aligned} \begin{aligned} E(U(q_{NC}; P_\gamma , \alpha , \sigma _N))= {P_N^\mu } q_{NC} - a {q_{NC}}^2 - \frac{F}{2} \end{aligned} \end{aligned}$$

(11)

$q_{NC}*$, the utility maximizing output level, is $\frac{{P_N^\mu }}{2a}$.

By substituting $q_{NC}*$ in (10) we derive expected utility of each farmer which is equal to $\frac{{{P_N^\mu }}^2}{4a} -\frac{F}{2})$.

1.2.3 Both farmers sell to private market

Farmer’s objective function when both of the sell everything to private market is:

$$\begin{aligned} \begin{aligned}&{\text {maximize}}&E(U(q_{NW}; P_\gamma , \alpha , \sigma _N)) = {P_N^\mu } q_{NW} - a{q_{NW}}^2 - \frac{\alpha }{2} \sigma {_N}^2 {q_{NW}}^2 \end{aligned} \end{aligned}$$

(12)

where $q_{NW}$ is the quantity produced and sold by each farmer in the private market.

The utility maximizing quantity for each farmer selling to private market is $\frac{{P_N^\mu }}{2a +\alpha \sigma {_N}^2}$.

And, the expected utility is equal to $\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$.

1.2.4 One farmer sells through cooperative and another in private market

When only one farmer participates in the cooperative, he/she incurs the entire operational cost. Therefore, the expected utility function is represented by the following Eq. (11):

$$\begin{aligned} \begin{aligned}&\underset{q_NC}{\text {maximize}}&E(U(q_{NC}; P_\gamma , \alpha , \sigma _N)) = {P_N^\mu } q_{NC} - a {q_{NC}}^2 - F \end{aligned} \end{aligned}$$

(13)

Expected utility maximizing output level:

$$\begin{aligned} q_{NC}* = \frac{{P_N^\mu }}{2a} \end{aligned}$$

(14)

Expected utility of the farmer from selling through cooperative:

$$\begin{aligned} E(U)^* = \frac{{{P_N^\mu }}^2}{4a} - F \end{aligned}$$

(15)

Expected utility from selling in private market is represented as follows:

$$E(U(q_{{NW}} ;\;P_{\gamma } ,\alpha ,\sigma _{N} )) = P_{N}^{\mu } q_{{NW}} - aq_{{NW}} ^{2} - \frac{\alpha }{2}\sigma {\text{ }}N^{2} q_{{NW}} ^{2}$$

(16)

The utility maximizing quantity level of farmer selling in private market is represented by $\frac{{P_N^\mu }}{2a +\alpha \sigma {_N}^2}$.

And, the expected utility is $\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$.

We represent these payoffs using a $2\times 2$ matrix:

	Cooperative	Private
Cooperative	$(\frac{{{P_N^\mu }}^2}{4a}-\frac{F}{2})$, $(\frac{{{P_N^\mu }}^2}{4a}-\frac{F}{2})$	$(\frac{{{P_N^\mu }}^2}{4a}-F)$, $\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$
Private	$\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$, $(\frac{{{P_N^\mu }}^2}{4a}-F)$	$\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$, $\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$

Nash equilibrium:

	Conditions	Nash Equilibrium
Case 1	$\frac{\alpha \sigma {_N}^2 {{P_N^\mu }}^2}{4a(2a+\alpha \sigma {_N}^2 )} \ge F$	(Cooperative, Cooperative)
Case 2	$\frac{\alpha \sigma {_N}^2 {{P_N^\mu }}^2}{4a(2a+\alpha \sigma {_N}^2 )} \le F \le \frac{\alpha \sigma {_N}^2 {{P_N^\mu }}^2}{2a(2a+\alpha \sigma {_N}^2 )}$	(Cooperative, Cooperative);(Private, Private)
Case 3	$\frac{\alpha \sigma {_N}^2 {{P_N^\mu }}^2}{2a(2a+\alpha \sigma {_N}^2 )} \le F$	(Private, Private)

1.3 Appendix 3

1.3.1 Both grow organic crop

The average price and the variance of organically produced crop are assumed to be ${P_O^\mu }$ and $\sigma {_O}^2$, respectively. The additional cost incurred by organic crop (in controlling for pesta and diseases), also called the cost of effort, is denoted by $C(\epsilon )= bq_{\gamma \delta }^2$.

1.3.2 Both participate in cooperative

The following equation represents the objective function of each farmer when both of them participates in a cooperative:

$$\begin{aligned} \begin{aligned}&\underset{q_OC}{\text {maximize}}&E(U(q_{OC}; P_\gamma , \alpha , \sigma _O)) = {P_O^\mu } q_{OC} - a{q_{OC}}^2 - b{q_{OC}}^2 - \frac{\alpha }{2} \sigma {_O}^2 {q_{OC}}^2 - \frac{F}{2} \end{aligned} \end{aligned}$$

(17)

where $q_{OC}$ is each farmer’s produce to be sold via cooperative.

By substituting $\sigma {_O}^2 = 0$ in Eq. (17)

$$\begin{aligned} \begin{aligned} E(U(q_{OC}; P_\gamma , \alpha , \sigma _O))= {P_O^\mu } q_{OC} - a {q_{OC}}^2- b{q_{OC}}^2 - \frac{F}{2} \end{aligned} \end{aligned}$$

(18)

The utility maximizing output level ‘$q_{OC}*$’ is $\frac{{P_O^\mu }}{2(a+b)}$ for each farmers.

Again, by substituting $q_{OC}*$ in the expected utility function, we get expected utility equal to $\frac{{{P_O^\mu }}^2}{4(a+b)} - \frac{F}{2}$.

1.3.3 Both sell in private market

The expected utility of each farmer:

$$\begin{aligned} \begin{aligned}&{\text {maximize}}&E(U(q_{OW}; P_\gamma , \alpha , \sigma _O)) = {P_O^\mu } q_{OW} - a{q_{OW}}^2 -b{q_{OW}}^2 - \frac{\alpha }{2} \sigma {_O}^2 {q_{OW}}^2 \end{aligned} \end{aligned}$$

(19)

where $q_{OW}$ is the quantity produced and sold by each farmer in the private market.

The expected utility maximizing output level of farmer selling in private market is represented by $\frac{{P_O^\mu }}{2a + 2b +\alpha \sigma {_O}^2}$.

Expected utility of each farmer equals to $\left( \frac{{{P_O^\mu }}^2}{2(2a + 2b + \alpha \sigma {_O}^2})\right)$.

1.3.4 One farmer sells in cooperative and another to private market

When there is only one farmer in the cooperative, he/she incurs the entire operational cost. Therefore, following Eq. (18), expected utility function of such farmer is:

$$\begin{aligned} \begin{aligned}&\underset{q_OC}{\text {maximize}}&E(U(q_{OC}; P_\gamma , \alpha , \sigma _O)) = {P_O^\mu } q_{OC} - a {q_{OC}}^2- b {q_{OC}}^2 - F \end{aligned} \end{aligned}$$

(20)

Expected utility maximizing output level:

$$\begin{aligned} q_{OC}*&= \frac{{P_O^\mu }}{2(a+b)} \end{aligned}$$

(21)

$$\begin{aligned} E(U)^*&= \frac{{{P_O^\mu }}^2}{4(a+b)} - F \end{aligned}$$

(22)

The expected utility of farmer selling in private market is represented by:

$$\begin{aligned}&E(U(q_{OW}; P_\gamma , \alpha , \sigma _O)) = {P_O^\mu } q_{OW} - a {q_{OW}}^2 - b{q_{OW}}^2 - \frac{\alpha }{2}\sigma {_O}^2 {q_{OW}}^2 \end{aligned}$$

(23)

$$\begin{aligned}&E(U^*) =\frac{{P_O^\mu }}{2a + 2b +\alpha \sigma {_O}^2} \end{aligned}$$

(24)

Expected utility received by each farmer equals to $\left( \frac{{{P_O^\mu }}^2}{2(2a + 2b + \alpha \sigma {_O}^2})\right)$.

We can represent these payoffs using a $2\times 2$ matrix:

	Cooperative	Private
Cooperative	$\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - \frac{F}{2}\right)$, $\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - \frac{F}{2}\right)$	$\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - F\right)$, $\left( \frac{{{P_O^\mu }}^2}{2(2a + 2b + \alpha \sigma {_O}^2)}\right)$
Private	$\left( \frac{{{P_O^\mu }}^2}{2(2a + 2b + \alpha \sigma {_O}^2)}\right)$, $\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - F\right)$	$\left( \frac{{{P_O^\mu }}^2}{2(2a + 2b + \alpha \sigma {_O}^2)}\right)$, $\left( \frac{{{P_O^\mu }}^2}{2(2a + 2b + \alpha \sigma {_O}^2)}\right)$

Nash equilibrium for the game:

	Conditions	Nash Equilibrium
Case 1	$\frac{\alpha \sigma {_O}^2 {{P_O^\mu }}^2}{4(a+b)(2a + 2b+\alpha \sigma {_O}^2)}\ge F$	(Cooperative, Cooperative)
Case 2	$\frac{\alpha \sigma {_O}^2 {{P_O^\mu }}^2}{4(a+b)(2a+2b+\alpha \sigma {_O}^2 )} \ge F \ge \frac{\alpha \sigma {_O}^2 {{P_O^\mu }}^2}{2(a+b)(2a+2b+\alpha \sigma {_O}^2)}$	(Cooperative, Cooperative);(Private, Private)
Case 3	$\frac{\alpha \sigma {_O}^2 {{P_O^\mu }}^2}{2(2a+2b+\alpha \sigma {_O}^2)} \le F$	(Private market, Private market)

1.4 Appendix 4

1.4.1 Choice of farming technique

Let $T^N_1 = \frac{\alpha \sigma {_N}^2 {{P_N^\mu }}^2}{4a(2a+\alpha \sigma {_N}^2 )}$ and $T^N_2 = 2T^N_1$.

$T^O_1 = \frac{\alpha \sigma {_O}^2 {{P_O^\mu }}^2}{4(a+b)(2a + 2b+\alpha \sigma {_O}^2)}$ and $T^O_2 = 2T^O_1$

Four possible situations arises:

(1) $T^N_1 < T^O_1$ and $T^N_2 < T^O_1$

(2) $T^N_1 < T^O_1$ and $T^N_2 > T^O_1$

(3) $T^N_1 > T^O_1$ and $T^N_1 > T^O_2$

(4) $T^N_1 > T^O_1$ and $T^N_1 < T^O_2$

However, $T^N_1 < T^O_1$ and $T^N_2 < T^O_1$ will hold true $\forall \alpha$ such that,

$$\begin{aligned} \alpha \ge \frac{4(a+b)^2 {P_N^\mu }^2 \sigma _N^2 - 2 a^2 {P_O^\mu }^2 \sigma _N^2}{(2(a+b){P_N^\mu }^2 - a\sigma _O^2)\sigma _N^2\sigma _O^2} \end{aligned}$$

(25)

and,

$$\begin{aligned} 0\le \alpha \le 1 \end{aligned}$$

(26)

The above scenarios represent five possible Nash equilibria for different ranges of“F”.

(a) $F \le T^N_1$: A range of very low F. Nash equilibrium is (Cooperation, Cooperation).

(b) $T^N_1 \ge F \le T^N_2$: low F. Nash equilibrium for non-organic crop is {(Cooperation, Cooperation); (Private, Private)}, whereas for organic crop is (Cooperation, Cooperation)

(C) $T^N_2 \ge F \le T^O_1$: moderate F. Nash equilibrium for non-organic crop is (Private, Private), whereas for organic crop is (Cooperation, Cooperation).

(d) $T^O_1 \ge F \le T^O_2$ i.e., high F. Nash equilibrium for non-organic crop is (Private, Private), whereas for organic crop is {(Cooperation, Cooperation); (Private, Private)}.

(e) $F \ge T^O_2$ i.e., very high F. Nash equilibrium is (Private, Private).

Case 1 $F \le T^N_1$ (Both participates in Cooperative)

	Organic(O)	Non-organic (N)
Organic(O)	$\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - \frac{F}{2}\right)$, $\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - \frac{F}{2}\right)$	$\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - F\right)$, $(\frac{{{P_N^\mu }}^2}{4a}- F)$
Non-organic (N)	$(\frac{{{P_N^\mu }}^2}{4a}-F)$, $\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - F\right)$	$(\frac{{{P_N^\mu }}^2}{4a}-\frac{F}{2})$, $(\frac{{{P_N^\mu }}^2}{4a}-\frac{F}{2})$

For any value of $\frac{{P_O^\mu }^2}{{P_N^\mu }^2}\ge \frac{(a+b)}{a}$ and $F\le \frac{a{P_O^\mu }^2-{P_N^\mu }^2(a+b)}{2(a+b)}$, a rational producer will always choose organic crop and sell to cooperative.

Case 2 $T^N_1<F<T^N_2$ (Non-organic Farmer sell through cooperative or private market and Organic Farmer through Cooperative)

The equilibrium for the game will be same as in case 1 if farmers producing non-organic crop sell through cooperative. But, the game changes as below if he sells through private market.

	Organic(O)	Non-organic (N)
Organic(O)	$\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - \frac{F}{2}\right)$, $\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - \frac{F}{2}\right)$	$\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - F\right)$, $\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$
Non-organic (N)	$\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$, $\left( \frac{{{P_O^\mu }}^2}{4(a+b)} - F\right)$	$\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$, $\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$

If $\frac{2a({P_O^\mu }^2-{P_N^\mu }^2)-2b{P_N^\mu }^2+\alpha \sigma _N^2{P_O^\mu }^2}{4(a+b)(2a+\alpha \sigma _N^2)} \ge F$, N.E. is both producing organic crop and selling in cooperative. If $\frac{2a({P_O^\mu }^2-{P_N^\mu }^2)-2b{P_N^\mu }^2+\alpha \sigma _N^2{P_O^\mu }^2}{4(a+b)(2a+\alpha \sigma _N^2)}\le F \le \frac{2a({P_O^\mu }^2-{P_N^\mu }^2)-2b{P_N^\mu }^2+\alpha \sigma _N^2{P_O^\mu }^2}{2(a+b)(2a+\alpha \sigma _N^2)}$, the equilibrium is either both producing organically and selling through cooperative or both producing non-organically and selling though Private market. Since both producing organic and selling in cooperative in pareto optimal, we assume farmers to produce organic and sell in cooperative

For $\frac{2a({P_O^\mu }^2-{P_N^\mu }^2)-2b{P_N^\mu }^2+\alpha \sigma _N^2{P_O^\mu }^2}{2(a+b)(2a+\alpha \sigma _N^2)}\le F$, the equilibrium will be both producing non-organically and selling through Private market.

Case 3 $T^N_2<F<T^O_1$ (Farmer producing non-organic sells to private market and Farmer producing organic sells to Cooperative)

Equilibrium is same as case 2.

Case 4 $T^O_1<F<T^O_2$ (Farmer producing non-organic sells to private market and farmer producing Organic sells to either Cooperative or Private market)

Nash equilibrium is same as Case 3 when farmer producing organic crop sells in cooperative.

However, the game matrix follows as below if he/she sells to private market:

	Organic(O)	Non-organic (N)
Organic(O)	$\left( \frac{{{P_O^\mu }}^2}{2(2a + 2b + \alpha \sigma {_O}^2)}\right)$, $\left( \frac{{{P_O^\mu }}^2}{2(2a + 2b + \alpha \sigma {_O}^2)}\right)$	$\left( \frac{{{P_O^\mu }}^2}{2(2a + 2b + \alpha \sigma {_O}^2)}\right)$, $\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$
Non-organic (N)	$\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$, $\left( \frac{{{P_O^\mu }}^2}{2(2a + 2b + \alpha \sigma {_O}^2)}\right)$	$\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$, $\frac{1}{2}\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$

If $\frac{{P_O^\mu }^2}{{P_N^\mu }^2} > \frac{2a+2b+ \alpha \sigma _O^2}{(2a+\alpha \sigma _N^2)}$, N.E. is both of them producing organic crop and selling to Private market. This condition will always hold true if $\frac{\sigma _N^2}{\sigma _O^2}\ge \frac{a}{(a+b)}$.

Case 5 $F \ge T^O_2$ (Both farmers selling through the Private market)

Equilibrium remains same as case 4.

The results of the entire game are summarized in Fig. 4.

Fig. 4

Cost of cooperative and Nash Equilibrium

1.5 Appendix 5

1.5.1 A comparative analysis

	Total quantity produced	Total utilities
$F \le \frac{2a({P_O^\mu }^2-{P_N^\mu }^2)-2b{P_N^\mu }^2+\alpha \sigma _N^2{P_O^\mu }^2}{2(a+b)(2a+\alpha \sigma _N^2)}$	$\frac{{P_O^\mu }}{(a+b)}$	$\left( \frac{{{P_O^\mu }}^2}{2(a+b)} - F\right)$
$\frac{2a({P_O^\mu }^2-{P_N^\mu }^2)-2b{P_N^\mu }^2+\alpha \sigma _N^2{P_O^\mu }^2}{2(a+b)(2a+\alpha \sigma _N^2)} \le F \le \frac{\alpha \sigma _O^2{P_O^\mu }^2}{4(a+b)(2a+2b+\alpha \sigma _O^2)}$	$\left( \frac{2{P_N^\mu }}{2a +\alpha \sigma {_N}^2}\right)$	$\left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$
$F \ge \frac{\alpha \sigma _O^2{P_O^\mu }^2}{4(a+b)(2a+2b+\alpha \sigma _O^2)}$	$\frac{2{P_O^\mu }}{2a + 2b +\alpha \sigma {_O}^2}$	$\left( \frac{{{P_O^\mu }}^2}{(2a + 2b + \alpha \sigma {_O}^2})\right)$

Let $A =\left( \frac{{{P_O^\mu }}^2}{2(a+b)} - F\right)$ , $B= \left( \frac{{{P_N^\mu }}^2}{2a +\alpha \sigma {_N}^2}\right)$ and $C=\left( \frac{{{P_O^\mu }}^2}{(2a + 2b + \alpha \sigma {_O}^2})\right)$

$$\begin{aligned} \text {Min A} = \frac{{P_N^\mu }^2}{2a+\alpha \sigma _N^2} \end{aligned}$$

(27)

Hence, $A\ge B$

For $a>0$, $b>0$ and $\frac{{P_N^\mu }^2}{{P_O^\mu }^2} \ge \frac{2 a +\alpha {\sigma _N^2}}{2 a+2 b+ \alpha {\sigma _O^2}}$, $A\ge B\ge C$.

Also, for $b<\frac{\alpha ({\sigma _N^2-\sigma _O^2})}{2}$, $\frac{{P_O^\mu }}{(a+b)}\ge \frac{2{P_N^\mu }}{2a +\alpha \sigma {_N}^2}\ge \frac{2{P_O^\mu }}{2a + 2b +\alpha \sigma {_O}^2}$

1.6 Appendix 6

See Table 7, 8.

Table 7 Region-wise number of cooperatives and organic and conventional transactions

Table 8 Product-wise organic and conventional transactions