Organic Mergers and Acquisitions

Abstract

This paper examines the competition between organic and non-organic firms, their incentives to undertake a horizontal merger, and the effect of mergers on firms’ market shares. We also consider an alternative setting where one firm can acquire its rival. For generality, we allow for product differentiation, demand, and cost asymmetries. Our results show that both organic and non-organic firms, despite their cost asymmetries and demand differentials, have incentives to merge under large conditions. When demand and cost differentials are significant, we identify settings under which a firm (either organic or non-organic) purchases its rival, to subsequently shut it down, and yet increase its profits. We then study under which conditions the merger can be welfare improving, which is more likely when goods are highly differentiated and their production costs are relatively symmetric.

Appendices

Appendix

Appendix 1 - Numerical Example

We evaluate cutoffs c₁ through c₆ at M= N = 5 firms throughout this Appendix. A similar analysis applies to settings with different numbers of firms.

Benchmark Case

Applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.79 \$and $c_{\text {NO}}^{M}=0.16$. Since c_O ≥ c_NO by definition, c_O ≥ 0.25. As a consequence, Region I cannot be sustained in equilibrium; Region II can be supported for all costs satisfying 0.25 ≤ c_O < 0.79; and Region III for all c_O ≥ 0.79. From Proposition 1, we obtain cutoffs c₁ through c₆ (see first row of Table 4). We can then conclude that: (1) the range of parameters [c₃,c₄] = [0.67, 0.99] is compatible with Region II as long as c_O ∈ [0.67, 0.79], which entails that this region can be supported in equilibrium when costs are relatively high and (2) the range of parameters [c₅,c₆] = [0.68, 0.97] is compatible for all values of c_O in Region III, implying that this region can be sustained for all admissible c_O.

We next examine the previous cutoffs on rows 2–5 of Table 1.

Lower Cost

c_NO = 1/10. Applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.71$ and $c_{\text {NO}}^{M}=-0.13$. Since c_O > 0 and c_O ≥ c_NO by definition, Region I cannot be sustained in equilibrium. Region II can be sustained for all cost satisfying 0.1 ≤ c_O < 0.71; and Region III for all c_O ≥ 0.71. Proposition 1 implies that: (1) the range of parameters [c₃,c₄] = [0.55, 0.99] is compatible with Region II as long as c_O ∈ [0.55, 0.71], which entails that this region can be supported in equilibrium when costs are moderately high and (2) the range of parameters [c₅,c₆] = [0.56, 0.96] is compatible for all values of c_O in Region III, entailing that this region can be sustained for all admissible c_O.

Higher Demand

a_NO = 1 . Applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.62$ and $c_{\text {NO}}^{M}=-0.5$. Since cutoff $c_{\text {NO}}^{M}<0$ and costs must be positive by definition, Region I cannot be sustained in equilibrium. Region II can be sustained for all costs satisfying 0.25 ≤ c_O < 0.62 and Region III for all remaining costs c_O ≥ 0.62. Proposition 1 implies that: (1) the range of parameters [c₃,c₄] = [0.41, 0.98] is compatible with Region II as long as c_O ∈ [0.42, 0.95] and (2) the range of parameters [c₅,c₆] = [0.42, 0.95] is compatible for all values of c_O in Region III, implying that this region can be sustained for all admissible c_O.

Homogeneous Goods

λ = 1. Applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=c_{\text {NO}}^{M}=0.58$. Since c_O ≥ c_NO by definition, c_O ≥ 0.25. As a consequence, Region I can be sustained for all costs satisfying 0.25 ≤ c_O < 0.58. Region II cannot be sustained since cutoffs coincide $c_{\text {NO}}^{M}=c_{O}={c_{O}^{M}}$ and Region III can be sustained for all c_O ≥ 0.58. Proposition 1 implies that: (1) the range of parameters [c₁,c₂] = [0.60, 0.69] is not compatible with Region I, implying that this region cannot be sustained in equilibrium; and (2) the range of parameters [c₅,c₆] = [0.59, 0.70] is compatible for all values of c_O in Region III, implying that this region can be sustained for all admissible c_O.

λ = 1 and a_NO = 1

Applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=c_{\text {NO}}^{M}=0.25$. Since c_O ≥ c_NO by definition, c_O ≥ 0.25. As a consequence, Region I cannot be sustained in equilibrium. Region II cannot be sustained since cutoffs ${c_{O}^{M}}$ and $c_{\text {NO}}^{M}$ coincide and Region III can be sustained for all c_O ≥ 0.25. Proposition 1 implies that the range of parameters [c₅,c₆] = [0.27, 0.47] is compatible for all values of c_O in Region III, entailing that this region can be sustained for all admissible c_O.

Appendix 2 - Extension to N Organic and M Non-Organic Firms

Second-Stage Output

Case 1, No Merger

If one or more firms chose to not merge during the first period, a merger does not occur, leading each firm to simultaneously and independently set its own output. In particular, every organic firm chooses its output level q_k to solve the following:

$$ \pi_{k}^{\text{NM}}\equiv \text{} \underset{q_{k}}{\max} \text{} (a_{k}-q_{k}-q_{-k}-\lambda Q_{j})q_{k}-c_{k}q_{k} $$

where $q_{-k}={\sum }_{i\neq j}^{N}q_{i}$ denotes the aggregate output by the other N − 1 organic firms, and $Q_{j}={\sum }_{r=1}^{M}q_{r}$ represents the aggregate output by all non-organic firms. Differentiating with respect to q_k, we obtain a best response function as follows:

$$ q_{k}(q_{-k},Q_{j})=\frac{1}{2}\left( a_{k}-c_{k}-q_{-k}-\lambda Q_{j}\right) $$

Invoking symmetry (q_k = q_i for every pair of organic firms k and i ), we find the following:

$$ q_{k}(Q_{j})=\frac{a_{k}-c_{k}-\lambda Q_{j}}{N+1}. $$

(4)

Solving a similar problem for every non-organic firm j, we have as follows:

$$ \pi_{j}^{NM}\equiv \text{} \underset{q_{j}}{\max} \text{} (a_{j}-q_{j}-q_{-j}-\lambda Q_{k})q_{j}-c_{j}q_{j} $$

which yields a symmetric best response function to q_k(q_−k,Q_j), that is,

$$ q_{j}(q_{-j},Q_{k})=\frac{1}{2} \left( a_{j}-c_{j}-q_{-j}-\lambda Q_{k}\right). $$

Invoking symmetry (q_j = q_r for every pair of non-organic firms j and r), we find the following:

$$ q_{j}(Q_{k})=\frac{a_{j}-c_{j}-\lambda Q_{k}}{M+1}. $$

(5)

In a symmetric equilibrium, aggregate output from organic firms is Q_k = Nq_k while that from non-organic firms is Q_j = Mq_j. Inserting these two properties in expressions (4) and (5), and simultaneously solving, yields the following:

$$ \begin{array}{@{}rcl@{}} q_{k}^{\text{NM}}&=&\frac{(M+1)(a_{k}-c_{k})-(a_{j}-c_{j})M\lambda} { (N+1)(M+1)-\text{MN}\lambda^{2}}\text{ \ \ \ and \ \ \ \ }\\ q_{j}^{\text{NM}}&=&\frac{ (N+1)(a_{j}-c_{j})-(a_{k}-c_{k})N\lambda} {(N+1)(M+1)-\text{MN}\lambda^{2}}. \end{array} $$

Last, equilibrium profits under no merger for the organic and non-organic firms are as follows:

$$ \pi_{k}^{\text{NM}}=\left[ \frac{(M+1)(a_{k}-c_{k})-(a_{j}-c_{j})M\lambda )}{ (N+1)(M+1)-\text{MN}\lambda^{2}}\right]^{2}=\left( q_{k}^{\text{NM}}\right)^{2} $$

and

$$ \pi_{j}^{\text{NM}}=\left[ \frac{(N+1)(a_{j}-c_{j})-(a_{k}-c_{k})N\lambda )}{ (N+1)(M+1)-\text{MN}\lambda^{2}}\right]^{2}=\left( q_{j}^{\text{NM}}\right)^{2}. $$

Firm k’s output under no merger is positive if its cost satisfies $c_{k} <c_{N,k}^{\text {NM}} \equiv \left (a_{k}-\frac {M \lambda a_{j}}{M+1}\right ) + \frac { M \lambda c_{j}}{M+1}$. Similarly firm j produces a positive output if only if $c_{j}<c_{M,j}^{\text {NM}} \equiv \left (a_{j}-\frac {N\lambda a_{k}}{N+1} \right ) +\frac {N\lambda c_{k}}{N+1}$.

Case 2, Merger

As in the model with only one firm of each type, we next examine each case separately.

Both Firms Are Active

When all types of firm are active, they maximize their joint profits as follows:

$$ \pi^{M,\text{Both}}\equiv \text{} \underset{q_{k},q_{j}}{\max} \text{} \left[ (a_{k}-q_{k}-q_{-k}-\lambda Q_{j})q_{k}-c_{k}q_{k}\right] +\left[ (a_{j}-q_{j}-q_{-j}-\lambda Q_{k})q_{j}-c_{j}q_{j}\right] $$

Differentiating with respect to q_k and q_j, and invoking symmetry for organic firms, q_−k = (N − 1)q_k, and for non-organic firms, q_−j = (M − 1)q_j, we obtain the following:

$$ \begin{array}{@{}rcl@{}} q_{k}^{M,\text{Both}}&=&\frac{2M(a_{k}-c_{k})-(a_{j}-c_{j})(M+N)\lambda}{ 4\text{MN}-(M+N)^{2}\lambda^{2}} \text{ \ \ \ and \ \ \ \ }\\ q_{j}^{M,\text{Both}}&=&\frac{ 2M(a_{j}-c_{j})-(a_{k}-c_{k})(M+N)\lambda}{4\text{MN}-(M+N)^{2}\lambda^{2}}. \end{array} $$

yielding equilibrium profits of the following:

$$ \pi_{k}^{M,\text{Both}} = \frac{N\left[(a_{j} - c_{j})(M + N)\lambda - 2M(a_{k} - c_{k}) \right]\left[(a_{j} - c_{j})(M - N)\lambda + ((M + N)\lambda^{2} - 2M)(a_{k} - c_{k}) \right]}{\left[4\text{MN} - (M + N)^{2}\lambda^{2}\right]^{2}} $$

and

$$ \pi_{j}^{M,\text{Both}} = \frac{M\left[(a_{k} - c_{k})(M + N)\lambda - 2N(a_{j} - c_{j}) \right]\left[(a_{j} - c_{j})((M + N)\lambda^{2} - 2N) - (a_{k} - c_{k})(M - N)\lambda \right]}{\left[4\text{MN} - (M + N)^{2}\lambda^{2}\right]^{2}}. $$

Firm k’s output under the merger is positive if its cost satisfies $c_{k} < c_{N,k}^{M} \equiv \left (a_{k}-\frac {(M+N) \lambda a_{j}}{2M}\right ) + \frac {(M+N) \lambda c_{j}}{2M}$.

Only Firm k Is Active

If the merged firm shuts down all non-organic firms j, its profit-maximization problem becomes the following:

$$ \pi_{k}^{M,k}\equiv \underset{q_{k}}{\text{} \max} \text{} (a_{k}-q_{k}-q_{-k})q_{k}-c_{k}q_{k}\text{.} $$

Differentiating with respect to q_k and invoking symmetry, q_−k = (N − 1)q_k, yields output as follows:

$$ q_{k}^{M,k}=\frac{a_{k}-c_{k}}{2N}, $$

with associated profits as follow:

$$ \pi_{k}^{M,k}=\frac{(a_{k}-c_{k})^{2}}{4N}. $$

Comparing the profits that the merged firm obtains from keeping producing both goods, π^M,Both, against those where only firm k remains active, $\pi _{k}^{M,k}$, we obtain the following lemma. For presentation purposes, recall that firm j’s output under the merger is positive if its cost c_j satisfies $c_{j}<a_{j}-\frac {(M+N)\lambda } {2N}(a_{k}-c_{k})$ which is equivalent to $c_{k}<c_{M,j}^{M}\equiv \left (a_{k}-\frac {2Na_{j}}{ (M+N)\lambda } \right ) +\frac {2Nc_{j}}{(M+N)\lambda }$.

As in the main body of the paper, our subsequent analysis considers firm k as the organic producer, firm O, and its rival j as the non-organic firm, NO.

Lemma 3

The following three regions can arise in the(c_NO,c_O)-quadrant:

• 1. Region I. Only firm Oproduces positive output if$c_{O}<c_{M,NO}^{M}$.

• 2. Region II. Both firms produce positive outputif$c_{M,NO}^{M}\leq c_{O}<c_{N,O}^{M}$.

• 3. Region III. Only firm NOproduces positive outputif$c_{O}\geq c_{N,O}^{M}$.

Proof of Lemma 3.

The profit difference $\pi ^{M,\text {Both}}-\pi _{O}^{M,O}$ yields a U-shaped curve, which becomes zero at exactly $c_{O}=c_{M,NO}^{M}\equiv \left (a_{O}-\frac {2Na_{NO}}{(M+N)\lambda } \right ) + \frac {2Nc_{NO}}{(M+N)\lambda }$ . As a consequence, $\pi ^{M,\text {Both}}\geq \pi _{O}^{M,O}$ holds for all parameter values. For the non-organic firm, the profit difference $\pi ^{M,\text {Both}}-\pi _{\text {NO}}^{M,\text {NO}}$ exhibits a similar shape, becoming zero at $c_{O}=c_{N,O}^{M}$, thus implying that $\pi ^{M,\text {Both}}\geq \pi _{\text {NO}}^{M,\text {NO}}$ also holds for all parameter values. Summarizing, it is profitable to maintain both firms active, rather than shutting one of them down.

We can now compare cutoffs $c_{N,O}^{M}$ and $c_{M,\text {NO}}^{M}$. First, cutoff $c_{N,O}^{M}$ originates above $c_{M,\text {NO}}^{M}$ since their vertical intercepts satisfy $a_{O}-\frac {(M+N)\lambda a_{\text {NO}}}{2M}>a_{O}-\frac {2Na_{\text {NO}}}{ (M+N)\lambda }$ , which holds given that λ ∈ [0, 1] by definition. Second, the positive slope of cutoff $c_{N,O}^{M}$is $\frac { (M+N)\lambda } {2M}$, whereas that of cutoff $c_{M,\text {NO}}^{M}$ is $\frac {2N}{ (M+N)\lambda }$ , thus indicating that cutoff $c_{M,\text {NO}}^{M}$ grows faster than $c_{N,O}^{M}$ does. In addition, cutoffs $c_{N,O}^{M}$ and $c_{M,\text {NO}}^{M}$ cross each other at c_NO = a_NO and a height of c_O = a_O. Recalling that a_O > c_O for every firm O, only three regions can be sustained in the (c_NO,c_O)-quadrant: (1) when $c_{O}<c_{M,NO}^{M}$, only firm O is active; (2) when $c_{M,NO}^{M}\leq c_{O}<c_{N,O}^{M}$, both firms are active; and (3) when $c_{O}\geq c_{N,O}^{M}$, only firm NO is active. □

We next examine how the results in Lemma 3 differ relative to its analogous result in the main body of the paper, Lemma 1.

Corollary 2

When the number of organic and non-organic firms coincide,M = N,cutoff$c_{N,O}^{M}$becomes${c_{O}^{M}}$andcutoff$c_{M,\text {NO}}^{M}$becomes$c_{\text {NO}}^{M}$.When the differenceM − Nincreases,both cutoffs$c_{M,\text {NO}}^{M}$and$c_{N,O}^{M}$shiftupwards, shrinking Regions II and III while expanding Region I.

Therefore, our results in Lemma 1 (which assumes one firm of each type) coincide with those where M firms of each type exist, regardless of how large the number of firms is, i.e., for all M = N ≥ 1. The findings depicted in Fig. 1 readily apply in this context.

However, as the difference in the number of non-organic and organic firms (as captured by M − N) increases, organic firms produce a positive output under more restrictive cost conditions. As a result, when the costs of organic firms are extremely high, as in Region III (see Fig. 1 in the main body of the paper), the merged firm produces non-organic goods alone. When the costs of both firms are relatively similar and high, as in Region II, they both remain active after the merger. In the rest of the cases, the merged firm produces organic goods alone as in Region I.

When products are homogeneous, λ = 1, our results in the main body of the paper still apply when the number of organic and non-organic firms coincides, M = N, as depicted in Fig. 2a. However, when M > N, cutoff $c_{N,O}^{M}$ originates below $c_{M,O}^{M}$, giving rise to Region II, where both firms are active; a result that could not be sustained when products are homogeneous and M = N. In contrast, when products are completely differentiated, λ = 0, all cutoffs identified in Lemma 3 simplify to a_O (i.e., they all become a horizontal line originating at a_O), yielding only one possible outcome in equilibrium: Region I, where the organic firm is the only active plant.

First Stage

For each (c_NO,c_O)-pair, every firm O anticipates the output profile that will emerge in the second stage of the game, i.e., Regions I–III. For completeness, we consider that during the first stage firms choose whether to merge, and subsequently examine how the results would change if, instead, one firm is allowed to acquire its rival.

In the first stage, every firm chooses whether to merge or not. In particular, for each regions I–III, the firm compares the profits that it currently obtains as an independent firm, $\pi _{N,O}^{\text {NM}}$, against the profits it would obtain under the merger: HCode $\frac {\pi _{N,O}^{M,O}}{M+N}$ in Region I, $\frac {\pi _{N,M}^{M,\text {Both}}}{M+N}$ in Region II, and $\frac {\pi _{M,NO}^{M,NO}}{M+N}$ in Region III. For simplicity, we assume that firms evenly share merger profits.

Proposition 3

During the first stage, every firm O chooses to merge as follows:

• 1. If (c_NO,c_O)-pairslie in Region I, firm O merges if and only ifc_O ∈ [c₁,c₂].

• 2. If (c_NO,c_O)-pairslie in Region II, firm O merges if and only ifc_O ∈ [c₃,c₄].

• 3. If (c_NO,c_O)-pairslie in Region III, firm O merges if and only ifc_O ∈ [c₅,c₆].

For compactness, cutoffsc₁ − c₆aredefined in the proof below.

Proof of Proposition 3.

First, consider (c_NO,c_O)-pairs in Region I, i.e., $c_{O}<c_{M,\text {NO}}^{M}$. In this region, only firm O operates under a merger in the second stage. Therefore, every firm O chooses to merge in the first stage if and only if its share of profits under the merger, $\frac {\pi _{N,O}^{M,O}}{M+N}$, exceeds the profits it would obtain as an independent firm, $\pi _{N,O}^{\text {NM}}$. Setting $\frac {\pi _{N,O}^{M,O}}{ M+N}\geq \pi _{N,O}^{\text {NM}}$, and solving for cost c_O, we find that $\frac { \pi _{N,O}^{M,O}}{M+N}\geq \pi _{N,O}^{\text {NM}}$ holds for all c_O ∈ [c₁,c₂], where,

$$ c_{1}\equiv \frac{(4a_{O}N(M+N)(M+1)^{2}-a_{O}\left( (N+1)(M+1)-M\lambda^{2}\right)^{2}}{4N(M+N)(M+1)^{2}-\left( (N+1)(M+1)-\text{MN}\lambda^{2}\right)^{2}} $$

$$ -\frac{4\text{MN}(M+N)(M+1)\lambda (a_{\text{NO}}-c_{\text{NO}})}{4N(M+N)(M+1)^{2}-\left( (N+1)(M+1)-\text{MN}\lambda^{2}\right)^{2}} $$

$$ -\frac{\frac{2M(a_{\text{NO}}-c_{\text{NO}})\lambda \left( (N+1)(M+1)-\text{MN}\lambda^{2}\right)} {\sqrt{N(M+N)}}}{4N(M+N)(M+1)^{2}-\left( (N+1)(M+1)-\text{MN}\lambda^{2}\right)^{2}}\text{.} $$

and

$$ c_{2}\equiv \frac{(4a_{O}N(M+N)(M+1)^{2}-a_{O}\left( (N+1)(M+1)-M\lambda^{2}\right)^{2}}{4N(M+N)(M+1)^{2}-\left( (N+1)(M+1)-MN\lambda^{2}\right)^{2}} $$

$$ -\frac{4\text{MN}(M+N)(M+1)\lambda (a_{\text{NO}}-c_{\text{NO}})}{4N(M+N)(M+1)^{2}-\left( (N+1)(M+1)-MN\lambda^{2}\right)^{2}} $$

$$ +\frac{\frac{2M(a_{NO}-c_{NO})\lambda \left( (N+1)(M+1)-MN\lambda^{2}\right)} {\sqrt{N(M+N)}}}{4N(M+N)(M+1)^{2}-\left( (N+1)(M+1)-\text{MN}\lambda^{2}\right)^{2}}\text{.} $$

Second, consider (c_NO,c_O)-pairs in Region II, i.e., $c_{M,\text {NO}}^{M}\leq c_{O}<c_{N,O}^{M}$. In this region, both firms are active under a merger in the second stage. Therefore, every firm O chooses to merge in the first stage if and only if its share of profits under the merger, $\frac {\pi _{N,M}^{M,\text {Both}}}{M+N}$, exceeds the profits it would obtain as an independent firm, $\pi _{N,O}^{\text {NM}}$. Setting $\frac { \pi _{N,M}^{M,\text {Both}}}{M+N}\geq \pi _{N,O}^{\text {NM}}$, and solving for cost c_O, we find that $\frac {\pi _{N,M}^{M,\text {Both}}}{M+N}\geq \pi _{N,O}^{\text {NM}}$ holds for all c_O ∈ [c₃,c₄], where,

$$ \begin{array}{@{}rcl@{}} c_{3}&\equiv& \frac{\frac{2a_{O}M-(a_{NO}-c_{\text{NO}})(M+N)\lambda}{ (M+N)(4\text{MN}-(M+N)^{2}\lambda^{2}}} {\left( \frac{2M}{(M+N)\left( 4\text{MN}-(M+N)^{2} \lambda^{2}\right)}-\frac{2(M+1)^{2}}{\left( (N+1)(M+1)-\text{MN}\lambda^{2} \right)^{2}}\right)}\\ &&-\frac{\frac{(a_{O}-c_{O})(M+1)\sqrt{1+N(2-4\text{MN}-3N^{2})}-\sqrt{ 2M(N-2(M+N)(M+N+\text{MN}))}\lambda+\text{MN}\lambda^{2}} {(M+N)\left( (N+1)(M+1)-\text{MN} \lambda^{2}\right)\sqrt{4\text{MN}-(M+N)^{2}\lambda^{2}}}} {\left( \frac{2M}{ (M+N)\left( 4\text{MN}-(M+N)^{2}\lambda^{2}\right)}-\frac{2(M+1)^{2}}{ \left( (N+1)(M+1)-\text{MN}\lambda^{2}\right)^{2}}\right)}\\ &&-\frac{\frac{2a_{O}(M+1)^{2}-2M(M+1)(a_{O}-c_{O})\lambda}{ \left( (N+1)(M+1)-\text{MN}\lambda^{2}\right)^{2}}} {\frac{2M}{(M+N) \left( 4\text{MN}-(M+N)^{2}\lambda^{2}\right)}-\frac{2(M+1)^{2}}{\left( (N+1)(M+1)-\text{MN} \lambda^{2}\right)^{2}}} \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} c_{4}&\equiv& \frac{\frac{4Ma_{O}-2(a_{NO}-c_{\text{NO}})(M+N)\lambda}{ (M+N)(4\text{MN}-(M+N)^{2}\lambda^{2})}} {\left( \frac{4M}{(M+N)\left( 4\text{MN}-(M+N)^{2} \lambda^{2}\right)}-\frac{4(M+1)^{2}}{\left( (N+1)(M+1)-\text{MN}\lambda^{2} \right)^{2}}\right)}\\ &&+\frac{\frac{2(a_{O}-c_{O})(M+1)\sqrt{1+N(2-4\text{MN}-3N^{2})}-\sqrt{ 2M(N-(2M+N)(M+N+\text{MN}))}\lambda+\text{MN}\lambda^{2}} {(M+N)\left( (N+1)(M+1)-\text{MN} \lambda^{2}\right)\sqrt{4\text{MN}-(M+N)^{2}\lambda^{2}}}} {\left( \frac{4M}{ (M+N)\left( 4\text{MN}-(M+N)^{2}\lambda^{2}\right)}-\frac{4(M+1)^{2}}{ \left( (N+1)(M+1)-\text{MN}\lambda^{2}\right)^{2}}\right)}\\ &&-\frac{\frac{4(M+1) \left( a_{O}(M+1)-(a_{NO}-c_{NO})M\lambda\right)}{ \left( (N+1)(M+1)-MN\lambda^{2}\right)^{2}}} {\left( \frac{4M}{ (M+N)\left( 4MN-(M+N)^{2}\lambda^{2}\right)}-\frac{4(M+1)^{2}}{ \left( (N+1)(M+1)-MN\lambda^{2}\right)^{2}}\right)} \end{array} $$

Third, consider (c_NO,c_O)-pairs in Region III, i.e., $c_{O}\geq c_{N,O}^{M}$. In this region, only firm NO operates under a merger in the second stage. Therefore, every firm O chooses to merge in the first stage if and only if its share of profits under the merger, $\frac {\pi _{M,\text {NO}}^{M,\text {NO}}}{M+N}$, exceeds the profits, it would obtain as an independent firm, $\pi _{N,O}^{\text {NM}}$. Setting $\frac {\pi _{M,\text {NO}}^{M,\text {NO}}}{M+N}\geq \pi _{N,O}^{\text {NM}}$, and solving for cost c_O, we find that this inequality holds for all c_O ∈ [c₅,c₆], where,

$$ \begin{array}{@{}rcl@{}} c_{5}&\equiv& a_{O}-\frac{(a_{\text{NO}}-c_{\text{NO}})M\lambda} {M+1}\\ &&-\frac{a_{\text{NO}}-c_{\text{NO}}}{\sqrt{M(M+N)}(N+1)(M+1)-\text{MN}\lambda^{2}}\\ &&\times\left( \frac{ (M+1)(N+1)^{2}}{2}+\frac{M^{2}\lambda^{2}}{2(M+1)}+\lambda^{2}-1\right) \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} c_{6}&\equiv& a_{O}-\frac{(a_{\text{NO}}-c_{\text{NO}})M\lambda} {M+1}\\ &&+\frac{a_{\text{NO}}-c_{\text{NO}}}{\sqrt{M(M+N)}(N+1)(M+1)-\text{MN}\lambda^{2}}\\ &&\times\left( \frac{ (M+1)(N+1)^{2}}{2}+\frac{M^{2}\lambda^{2}}{2(M+1)}+\lambda^{2}-1\right) \text{.} \end{array} $$

□

Numerical Example

Benchmark Case

We first consider the case when the number of organic and non-organic firms are equal. Applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.79 \$ and $c_{\text {NO}}^{M}=0.16$. Since c_O ≥ c_NO by definition, c_O ≥ 0.25. As a consequence, Region I cannot be sustained in equilibrium; Region II can be supported for all 0.25 ≤ c_O < 0.79; and Region III for all c_O ≥ 0.79. From Proposition 1, we obtain cutoffs c₁ through c₆ (see first row of Table 1). We can then conclude that (1) the range of parameters [c₃,c₄] = [0.67, 0.99] is compatible with Region II as long as c_O ∈ [0.67, 0.79], which entails that this region can be supported in equilibrium when costs are relatively high; and (2) the range of parameters [c₅,c₆] = [0.68, 0.97] is compatible for all values of c_O in Region III, implying that this region can be sustained for all admissible c_O.

Then, for the case when the number of non-organic firms are greater than that of organic firms, applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.84$ and $c_{\text {NO}}^{M}=0.44$. Since c_O ≥ c_NO by definition, c_O ≥ 0.25. As a consequence, Region I can only be sustained for all 0.25 ≤ c_O < 0.44; Region II can be supported for all 0.44 ≤ c_O < 0.84; and Region III for all c_O ≥ 0.84. We can then conclude that (1) the range of parameters [c₁,c₂] = [0.73, 0.85] is incompatible with Region I, and thus this region cannot be sustained in equilibrium; (2) the range of parameters [c₃,c₄] = [0.71, 0.89] is compatible with Region II as long as c_O ∈ [0.71, 0.84], which entails that this region can be supported in equilibrium when costs are relatively high; and (3) the range of parameters [c₅,c₆] = [0.72, 0.89] is compatible for all values of c_O in Region III , implying that this region can be sustained for all admissible c_O.

We next examine the previous cutoffs on rows 2–5 of Table 1.

Lower Cost

c_NO = 1/10. For the case when the number of organic and non-organic firms are equal, applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.71$ and $c_{\text {NO}}^{M}=-0.13$. Since c_O > 0 and c_O ≥ c_NO by definition, Region I cannot be sustained in equilibrium. Region II can be sustained for all costs satisfying 0.1 ≤ c_O < 0.71 and Region III for all c_O ≥ 0.71. Proposition 1 implies that (1) the range of parameters [c₃,c₄] = [0.55, 0.99] is compatible with Region II for all costs c_O ∈ [0.55, 0.71], which entails that this region can be supported in equilibrium when costs are moderately high and (2) the range of parameters [c₅,c₆] = [0.56, 0.96] is compatible for all values of c_O in Region III, entailing that this region can be sustained for all admissible c_O.

For the second case when the number of non-organic firms are greater than that of organic firms, applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.78$ and $c_{\text {NO}}^{M}=0.24$. Since c_O ≥ c_NO by definition, c_O ≥ 0.1. As a consequence, Region I can be sustained for all costs satisfying 0.1 ≤ c_O < 0.24; Region II can be supported for all 0.24 ≤ c_O < 0.78; and Region III for all c_O ≥ 0.78. Proposition 1 implies that (1) the range of parameters [c₁,c₂] = [0.64, 0.79] is incompatible with Region I, and thus this region cannot be sustained in equilibrium; (2) the range of parameters [c₃,c₄] = [0.61, 0.857] is compatible with Region II as long as c_O ∈ [0.61, 0.78], which entails that this region can be supported in equilibrium when costs are relatively high; and (3) the range of parameters [c₅,c₆] = [0.62, 0.854] is compatible for all values of c_O in Region III, entailing that this region can be sustained for all admissible c_O.

Higher Demand

a_NO = 1 . For the case when the number of organic and non-organic firms are equal, applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.62$ and $c_{\text {NO}}^{M}=-0.5$. Since cutoff $c_{\text {NO}}^{M}<0$ and costs must be positive by definition, Region I cannot be sustained in equilibrium. Region II can be sustained for all costs satisfying 0.25 ≤ c_O < 0.62 and Region III for all remaining costs c_O ≥ 0.62. Proposition 1 implies that (1) the range of parameters [c₃,c₄] = [0.41, 0.98] is compatible with Region II as long as c_O ∈ [0.41, 0.62] and (2) the range of parameters [c₅,c₆] = [0.42, 0.95] is compatible for all values of c_O in Region III, implying that this region can be sustained for all admissible c_O.

For the second case when the number of non-organic firms are greater than that of organic firms, applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.71$ and $c_{\text {NO}}^{M}=0$. Since costs must be positive by definition, Region I cannot be sustained in equilibrium. Region II can be sustained for all costs satisfying 0.25 ≤ c_O < 0.71 and Region III for all remaining costs c_O ≥ 0.71. Proposition 1 implies that (1) the range of parameters [c₃,c₄] = [0.49, 0.81] is compatible with Region II as long as c_O ∈ [0.49, 0.71] and (2) the range of parameters [c₅,c₆] = [0.51, 0.80] is compatible for all values of c_O in Region III, implying that this region can be sustained for all admissible c_O.

Homogeneous Goods

λ = 1. For the case when the number of organic and non-organic firms are equal, applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=c_{\text {NO}}^{M}=0.58$. Since c_O ≥ c_NO by definition, c_O ≥ 0.25. As a consequence, Region I can be sustained for all costs satisfying 0.25 ≤ c_O < 0.58. Region II cannot be sustained since cutoffs coincide $c_{\text {NO}}^{M}=c_{O}={c_{O}^{M}}$ and Region III can be sustained for all c_O ≥ 0.58. Proposition 1 implies that (1) the range of parameters [c₁,c₂] = [0.60, 0.69] is not compatible in Region I, implying that this region cannot be sustained in equilibrium and (2) the range of parameters [c₅,c₆] = [0.59, 0.70] is compatible for all values of c_O in Region III, implying that this region can be sustained for all admissible c_O.

For the second case when the number of non-organic firms are greater than that of organic firms, applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.68$ and $c_{\text {NO}}^{M}=0.72$. Since c_O ≥ c_NO by definition, c_O ≥ 0.25. As a consequence, Region I can be sustained for all cost satisfying 0.25 ≤ c_O < 0.72. Region II can be supported for all costs satisfying 0.68 ≤ c_O < 0.72 and Region III can be sustained for all c_O ≥ 0.68. Proposition 1 implies that (1) the range of parameters [c₁,c₂] = [0.58, 0.65] is compatible in Region I as long as c_O ∈ [0.58, 0.72], implying that this region can be sustained in equilibrium; (2) the range of parameters [c₃,c₄] = [0.60, 0.64] is not compatible with Region II; and (3) the range of parameters [c₅,c₆] = [0.59, 0.64] is not compatible in Region III, implying that this region cannot be sustained in equilibrium.

λ = 1 and a_NO = 1

For the case when the number of organic and non-organic firms are equal, applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=c_{\text {NO}}^{M}=0.25$. Since c_O ≥ c_NO by definition, c_O ≥ 0.25. As a consequence, Region I cannot be sustained in equilibrium, Region II cannot be sustained since cutoffs ${c_{O}^{M}}$ and $c_{\text {NO}}^{M}$ coincide, and Region III can be sustained for all c_O ≥ 0.25. Proposition 1 implies that the range of parameters [c₅,c₆] = [0.27, 0.47] is compatible for all values of c_O in Region III, entailing that this region can be sustained for all admissible c_O.

For the second case when the number of non-organic firms are greater than that of organic firms, applying Lemma 1 to this parametric example yields cutoffs ${c_{O}^{M}}=0.43$ and $c_{\text {NO}}^{M}=0.5$. Since c_O ≥ c_NO by definition, c_O ≥ 0.25. As a consequence, Region I can only be sustained for costs satisfying 0.25 ≤ c_O < 0.5. Region II can be supported for all costs satisfying 0.43 ≤ c_O < 0.5 and Region III can be sustained for all c_O ≥ 0.5. Proposition 1 implies that (1) the range of parameters [c₁,c₂] = [0.25, 0.37] is compatible in Region I, implying that this region can be sustained in equilibrium; (2) the range of parameters [c₃,c₄] = [0.29, 0.35] is not compatible with Region II; and (3) the range of parameters [c₅,c₆] = [0.27, 0.36] is not compatible in Region III, implying that this region cannot be sustained in equilibrium.

Proof of Lemma 1

First, note that the profit difference $\pi ^{M,\text {Both}}-\pi _{O}^{M,O}$ yields a U-shaped curve, which becomes zero at exactly $c_{O}=a_{O}-\frac { a_{\text {NO}}-c_{\text {NO}}}{\lambda }$. As a consequence, $\pi ^{M,\text {Both}}\geq \pi _{O}^{M,O}$ holds for all parameter values. For the non-organic firm, the profit difference $\pi ^{M,\text {Both}}-\pi _{\text {NO}}^{M,\text {NO}}$ exhibits a similar shape, becoming zero at $c_{O}={c_{O}^{M}}$, thus implying that $\pi ^{M,\text {Both}}\geq \pi _{\text {NO}}^{M,\text {NO}}$ also holds for all parameter values. Summarizing, it is profitable to maintain both firms active, rather than shutting one of them down. However, conditions $c_{O}<{c_{O}^{M}}$ and $c_{\text {NO}}<c_{\text {NO}}^{M}$ still apply yielding different regions in the (c_NO,c_O)-quadrant.

We can now compare cutoffs ${c_{O}^{M}}$ and $c_{\text {NO}}^{M}$. First, cutoff ${c_{O}^{M}}$ originates above $c_{\text {NO}}^{M}$ since their vertical intercepts satisfy $a_{O}-\lambda a_{\text {NO}}>a_{O}-\frac {a_{\text {NO}}}{\lambda }$, which holds given that λ ∈ [0, 1] by definition. Second, the positive slope of cutoff ${c_{O}^{M}}$is λ, whereas that of cutoff $c_{\text {NO}}^{M}$ is 1/λ, thus indicating that cutoff $c_{\text {NO}}^{M}$ grows faster than ${c_{O}^{M}}$ does. In addition, cutoffs ${c_{O}^{M}}$ and $c_{\text {NO}}^{M}$ cross each other at c_NO = a_NO and a height of c_O = a_O. Recalling that a_O > c_O for the organic firm, only three regions can be sustained in the (c_NO,c_O)-quadrant: (1) when $c_{O}<c_{\text {NO}}^{M}$, only the organic firm is active; (2) when $c_{\text {NO}}^{M}\leq c_{O}<{c_{O}^{M}}$, both firms are active; and (3) when $c_{O}\geq {c_{O}^{M}}$, only the non-organic firm is active.

Proof of Proposition 1

First, consider (c_NO,c_O)-pairs in Region I, i.e., $c_{O}<c_{\text {NO}}^{M}$ . In this region, only the organic firm operates under a merger in the second stage. Therefore, the organic firm chooses to merge in the first stage if and only if its share of profits under the merger, $\frac {\pi _{O}^{M,O}}{2}$, exceeds the profits, it would obtain as an independent firm, $\pi _{O}^{\text {NM}}$. Setting $\frac {\pi _{O}^{M,O}}{2}\geq \pi _{O}^{\text {NM}}$, and solving for cost c_O, we find that $\frac {\pi _{O}^{M,O}}{2}\geq \pi _{O}^{\text {NM}}$ holds for all c_O ∈ [c₁,c₂], where,

$$ c_{1}\equiv \frac{a_{O}(16+8\lambda^{2}-\lambda^{4})-16\lambda (a_{\text{NO}}-c_{\text{NO}})-2\sqrt{2}(a_{\text{NO}}-c_{\text{NO}})\lambda (4-\lambda^{2})}{ 16+8\lambda^{2}-\lambda^{4}} $$

and

$$ c_{2}\equiv \frac{a_{O}(16+8\lambda^{2}-\lambda^{4})-16\lambda (a_{\text{NO}}-c_{\text{NO}})+2\sqrt{2}(a_{\text{NO}}-c_{\text{NO}})\lambda (4-\lambda^{2})}{ 16+8\lambda^{2}-\lambda^{4}}\text{.} $$

Second, consider (c_NO,c_O)-pairs in Region II, i.e., $c_{\text {NO}}^{M}\leq c_{O}<{c_{O}^{M}}$. In this region, both firms are active under a merger in the second stage. Therefore, the organic firm chooses to merge in the first stage if and only if its share of profits under the merger, $\frac {\pi ^{M,\text {Both}}}{2}$, exceeds the profits, it would obtain as an independent firm, $\pi _{O}^{\text {NM}}$. Setting $\frac {\pi ^{M,\text {Both}}}{2}\geq \pi _{O}^{\text {NM}}$, and solving for cost c_O, we find that $\frac {\pi ^{M,\text {Both}}}{2}\geq \pi _{O}^{\text {NM}}$ holds for all c_O ∈ [c₃,c₄], where,

$$ c_{3}\!\equiv\! \frac{(a_{\text{NO}} - c_{\text{NO}})\lambda^{3}(8 + \lambda^{2}) + a_{O}(16 - 24\lambda^{2} - \lambda^{4}) + (a_{\text{NO}} - c_{\text{NO}})(4 - \lambda^{2}) \left[ 16 - 32\lambda^{2} + 15\lambda^{4} + \lambda^{6}\right]^{1/2}}{ 16 - 24\lambda^{2} - \lambda^{4}} $$

and

$$ c_{4}\!\equiv\! \frac{(a_{\text{NO}} - c_{\text{NO}})\lambda^{3}(8 + \lambda^{2}) + a_{O}(16 - 24\lambda^{2} - \lambda^{4}) - (a_{\text{NO}} - c_{\text{NO}})(4 - \lambda^{2}) \left[ 16 - 32\lambda^{2} + 15\lambda^{4} + \lambda^{6}\right]^{1/2}}{ 16 - 24\lambda^{2} - \lambda^{4}}\text{.} $$

Third, consider (c_NO,c_O)-pairs in Region III, i.e., $c_{O}\geq {c_{O}^{M}}$. In this region, only the non-organic firm operates under a merger in the second stage. Therefore, the organic firm chooses to merge in the first stage if and only if its share of profits under the merger, $\frac { \pi _{NO}^{M,\text {NO}}}{2}$, exceeds the profits it would obtain as an independent firm, $\pi _{O}^{\text {NM}}$. Setting $\frac {\pi _{\text {NO}}^{M,\text {NO}}}{2}\geq \pi _{O}^{\text {NM}}$ , and solving for cost c_O, we find that $\frac {\pi _{\text {NO}}^{M,\text {NO}}}{2}\geq \pi _{O}^{\text {NM}}$ holds for all c_O ∈ [c₅,c₆], where,

$$ c_{5}\equiv \frac{4[2a_{O}-\lambda (a_{\text{NO}}+c_{\text{NO}})]+\sqrt{2} (a_{\text{NO}}-c_{\text{NO}})(4-\lambda^{2})}{8} $$

and

$$ c_{6}\equiv \frac{4[2a_{O}-\lambda (a_{NO}+c_{NO})]-\sqrt{2} (a_{NO}-c_{NO})(4-\lambda^{2})}{8}\text{.} $$

Proof of Lemma 2

As discussed in the main body of the paper, social welfare is given by the sum of consumer and producer surplus, W = CS + PS, where $\text {CS}\equiv \frac {1}{2} ({q_{O}^{2}}+q_{\text {NO}}^{2}+2\lambda q_{O}q_{\text {NO}})$ and PS ≡ [p(q_O,q_NO)q_O − c_Oq_O] + [p(q_NO,q_O)q_NO − c_NOq_NO]. Using the inverse demands p(q_O,q_NO) and p(q_NO,q_O), the expression of producer surplus, PS, collapses to PS = a_Oq_O + q_NO(a_NO − c_NO − q_NO) − q_O(q_O + c_O + 2λq_NO). Differentiating welfare W with respect to q_O, we obtain the following:

$$ a_{O}-c_{O}-q_{O}-\lambda q_{\text{NO}}=0 $$

and a symmetric expression when we differentiate W with respect to q_NO , a_NO − c_NO − q_NO − λq_O = 0. Simultaneously solving for q_O and q_NO, yields $q_{O}^{\text {SO}}=\frac {a_{O}-c_{O}-\lambda (a_{\text {NO}}-c_{\text {NO}})}{ 1-\lambda ^{2}}$ and $q_{\text {NO}}^{\text {SO}}=\frac {a_{\text {NO}}-c_{\text {NO}}-\lambda (a_{O}-c_{O})}{ 1-\lambda ^{2}}$.

Comparing $q_{O}^{\text {SO}}$ against $q_{O}^{M,\text {Both}}$, we obtain that $q_{O}^{\text {SO}}\geq q_{O}^{M,\text {Both}}$ for all $c_{O}\leq {c_{O}^{A}}\equiv \left (a_{O}-\lambda a_{\text {NO}}\right ) +\lambda c_{\text {NO}}$, where the term in parenthesis indicates the vertical intercept of cutoff ${c_{O}^{A}}$ in the (c_NO,c_O)-quadrant (see Fig. 1 for a reference), while λ represents its positive slope. Similarly, comparing $q_{O}^{\text {SO}}$ against $q_{O}^{M,O}$, we obtain that $q_{O}^{\text {SO}}\geq q_{O}^{M,O}$ for all $c_{O}\leq {c_{O}^{B}}\equiv \left (a_{O}-\frac {2\lambda a_{\text {NO}}}{1+\lambda ^{2}}\right ) + \frac {2\lambda } {1+\lambda ^{2}}c_{\text {NO}}$, where the term in parenthesis indicates the vertical intercept of cutoff ${c_{O}^{B}}$ in the (c_NO,c_O)-quadrant (such as that in Fig. 1), while $\frac {2\lambda } { 1+\lambda ^{2}}$ represents its positive slope. A similar argument for the output levels of the non-organic firm yields that $q_{\text {NO}}^{\text {SO}}\geq q_{\text {NO}}^{M,\text {NO}}$ if and only if $c_{O}\leq \left (a_{O}-\frac {\left (1+\lambda ^{2}\right ) a_{\text {NO}}}{2\lambda } \right ) +\frac {\left (1+\lambda ^{2}\right ) c_{\text {NO}}}{2\lambda } \equiv c_{\text {NO}}^{B}$.

Proof of Proposition 2

Let us now analyze each of the regions in Fig. 1. Although we are simultaneously plotting several cutoffs for c_O, their ranking does not depend on λ.

As depicted in Fig. 7, cutoffs ${c_{O}^{M}},{c_{O}^{A}}$, and $c_{O}^{\text {SO}}$ coincide and originate above cutoff ${c_{O}^{B}}$ since $a_{O}-\lambda a_{\text {NO}}>a_{O}-\frac {2\lambda a_{\text {NO}}}{1+\lambda ^{2}}$, cutoff ${c_{O}^{B}}$ originates above $c_{\text {NO}}^{B}$ since $a_{O}-\frac {2\lambda a_{\text {NO}}}{1+\lambda ^{2}}>a_{O}-\frac {\left (1+\lambda ^{2}\right ) a_{\text {NO}}}{2\lambda }$, and cutoff $c_{\text {NO}}^{B}$ originates above $c_{\text {NO}}^{M}$ since $a_{O}-\frac {\left (1+\lambda ^{2}\right ) a_{\text {NO}}}{2\lambda } >a_{O}-\frac {a_{\text {NO}}}{\lambda }$ . Furthermore, cutoffs ${c_{O}^{M}}$ and $c_{O}^{\text {SO}}$ coincide and originate above cutoffs $c_{\text {NO}}^{M}$ and $c_{\text {NO}}^{\text {SO}}$ since $a_{O}-\lambda a_{\text {NO}}>a_{O}-\frac {a_{\text {NO}}}{\lambda }$.

Fig. 7

(in the Appendix)

Starting at Region I, where $c_{O}<c_{\text {NO}}^{M}$, only the organic firm is active in equilibrium. The social planner would also have only the organic firm being active since in this region c_O satisfies $c_{O}<c_{O}^{\text {SO}}$ and $c_{O}<c_{\text {NO}}^{\text {SO}}$. We can then compare equilibrium and socially optimal output, $q_{O}^{M,O}$ and $q_{O}^{\text {SO}}$, obtaining that $q_{O}^{\text {SO}}\geq q_{O}^{M,O}$ since Region I lies entirely below cutoff ${c_{O}^{B}}$. Therefore, a socially insufficient output emerges in Region I, relative to the social optimum.

In Region II, where $c_{\text {NO}}^{M}\leq c_{O}< {c_{O}^{M}}$, both firms are active in equilibrium where $q_{O}^{M,\text {Both}},q_{\text {NO}}^{M,\text {Both}}>0$. The social planner would also recommend that both firms are active since $c_{\text {NO}}^{\text {SO}}\leq c_{O}< c_{O}^{\text {SO}}$. Comparing equilibrium and optimal output, we obtain that $q_{O}^{\text {SO}}\geq q_{O}^{M,\text {Both}}$ since $c_{O}\leq {c_{O}^{A}}$. Therefore, a socially insufficient output emerges in Region II.

In Region III, only the non-organic firm is active in equilibrium, which coincides with the social optimum since $c_{O}\geq c_{O}^{\text {SO}}$ and $c_{O}\geq c_{\text {NO}}^{\text {SO}}$. Comparing output levels, we find that $q_{\text {NO}}^{\text {SO}}<q_{\text {NO}}^{M,\text {NO}}$ since Region III lies entirely above cutoff $c_{\text {NO}}^{B}$. Therefore, a socially excessive output emerges for all costs in Region III.

In summary, socially insufficient production occurs in two regions: (a) Region I, where $c_{O}<c_{\text {NO}}^{M}=c_{\text {NO}}^{\text {SO}}$; (b) Region II, where $c_{\text {NO}}^{M}\leq c_{O}<{c_{O}^{M}}=c_{O}^{\text {SO}}$. Furthermore, we can collapse the conditions to just $c_{O}<c_{O}^{\text {SO}}$. Socially excessive production occurs in Region III, where $c_{O}\geq {c_{O}^{M}}=c_{O}^{\text {SO}}$.

Proof of Corollary 1

Undifferentiated Products

λ = 1. In this context, all cutoffs collapse to the same line (i.e., they are all originating at a_O − a_NO), dividing into two regions, Region I and Region III, as depicted in Fig. 8.

Fig. 8

(in the Appendix): Cutoffs when λ = 1

Starting at Region I, where $c_{O}<c_{\text {NO}}^{M}$, only the organic firm is active in equilibrium. The social planner would also have only the organic firm being active since in this region c_O satisfies $c_{O}<c_{O}^{\text {SO}}$ and $c_{O}<c_{\text {NO}}^{\text {SO}}$. Comparing equilibrium and socially optimal output, we obtain that $q_{O}^{\text {SO}}\geq q_{O}^{M,O}$ since Region I lies entirely below cutoff ${c_{O}^{B}}$. As a result, a socially insufficient output emerges in Region I.

In Region III, only the non-organic firm is active in equilibrium. The social planner would also have only the non-organic firm being active since c_O satisfies $c_{O}\geq c_{\text {NO}}^{\text {SO}}$. Comparing output levels, we find that $q_{\text {NO}}^{\text {SO}}<q_{\text {NO}}^{M,\text {NO}}$ since Region III lies entirely above $c_{\text {NO}}^{B}$. Therefore, a socially excessive output emerges in Region III.

Completely Differentiated Products

λ = 0. As depicted in Fig. 9, in this setting, all cutoffs collapse to the same line (i.e., they are all horizontal line originating at a_O), yielding only one possible outcome: Region I, where the organic firm is the only active plant if $c_{O}<c_{\text {NO}}^{M}$.

Fig. 9

(in the Appendix): Cutoffs when λ = 0

In Region I, where $c_{O}<c_{\text {NO}}^{M}$, only the organic firm is active in equilibrium. The social planner would also have only the organic firm being active since in this region c_O satisfies $c_{O}<c_{O}^{\text {SO}}$ and $c_{O}<c_{\text {NO}}^{\text {SO}}$. We can then compare equilibrium and socially optimal output, $q_{O}^{M,O}$ and $q_{O}^{\text {SO}}$, obtaining that $q_{O}^{\text {SO}}\geq q_{O}^{M,O}$ since Region I lies entirely below cutoff ${c_{O}^{B}}$. Therefore, a socially insufficient output emerges in Region I, relative to the social optimum.