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Endogenous Equity Shares in Cournot Competition: Welfare Analysis and Policy

  • Kiriti Kanjilal and Félix Muñoz-García EMAIL logo


We consider a duopoly in which firms can strategically choose equity shares on their rival’s profits before competing in quantities. We identify equilibrium equity shares, and subsequently compare them against the optimal equity shares that maximize social welfare. Most previous studies assume that equity shares are exogenous, and those allowing for endogenous shares do not evaluate if equilibrium shares are socially excessive or insufficient. Our results also help us identify taxes on equity acquisition that induce firms to produce a socially optimal output without the need to directly tax output levels.

JEL Classification: L10; L2; D43; D6


We thank the editor and one anonymous referee for their insightful suggestions. We also appreciate the discussions and recommendations of Ana Espinola-Arredondo, Jinhui Bai, Raymond Batina, and Alan Love.


A Weakly concave cost of equity

In this appendix, we show that when β ≤ 1, firm i acquires no equity on firm j, αi=0, if (ac)2(1αj)(3αj)2b12βNWjF or maximal equity, αi=12, otherwise. Therefore, the firm’s equilibrium equity acquisition is at a corner solution (αi=0 or αi=12) if the cost of acquiring equity is weakly concave, that is, acquiring further shares becomes relatively cheaper as the firm owns a larger equity on its rival. Otherwise, the firm acquires a positive amount of equity as long as its cost is not extremely low or high. We prove this result below.


When β ≤ 1, the marginal cost of acquiring equity satisfies


Since β ≤ 1, the above expression satisfies 2MCαi20. The second-order condition yields 2MBαi22MCαi2. In addition, the marginal benefit from acquiring equity is increasing since 2MBαi2=24(1αi)(ac)2b(3αiαj)5>0, implying that


Therefore, the value of αi that solves MBαi=MCαi is a minimum and not a maximum. In such a case, the optimal α* is given by a corner solution, i. e. α* = 0 or α=12. The profits for firm i when acquiring no equity, αi = 0, are given by,


while the profits for firm i when acquiring maximal equity, αi=12, are


Summarizing, when β ≤ 1, the firm chooses no equity, αi=0, if


holds, while otherwise it chooses maximal equity, αi=12.   □

B Extension to convex production costs

In this appendix, we explore how our results change when firms face a convex production cost C(qi)=cqi2, where c > 0. For simplicity, we assume a = b = 1. In this context, firm i’s profit πi is


Equilibrium output – Second stage. Using this definition of πi in problem (2), and differentiating with respect to qi, we obtain best response function


where the vertical axis is 12(1+c) thus being unaffected by firms’ equity shares. Like in the main body of the paper, when firms hold no equity shares, αi=αj=0, the best response function collapses to 12(1+c)12(1+c)qj. When only firm i holds equity shares on firm j’s profits, αi > 0 but αj = 0, the best response function pivots inwards becoming 12(1+c)(1+αi)2(1+c)qj. Finally, when both firms sustain positive equity shares, αi,αj>0, the best response function pivots inwards even further.

Simultaneously solving for output levels qi and qj, we find


which is positive under all parameter values.

Equilibrium equity – First stage. We now substitute the solution for qi and qj into the profit function of firm i, πi, and obtain πi(αi,αj), which represents the profit that firm i earns during the second stage as a function of equity shares αi and αj. We can now insert profit πi(αi,αj) into firm i’s equity choice in the first-period game, as follows.


where, similarly as in expression (3), Cαi=δαi2 captures the cost of acquiring equity. Differentiating the above expression with respect to equity share αi to obtain firm i’s marginal benefit of acquiring equity, MBi. This marginal benefit is, however, very large in this setting of convex production costs. For tractability, we do not provide the expression of MBi here. However, we set MBi=MCi, and Table 6 below provides an analogous simulation of optimal equity share αi as in Table 1 of the paper using the same parameter values.

Table 6:

Optimal equity share αi with convex production costs.

Net worth NW/Marginal cost cc = 0c = 0.1c = 0.3c = 0.5c = 0.7c = 0.9
NW =
NW =
NW =
NW =
NW =
NW =
NW =
NW =
NW =
Table 7:

Optimal equity subsidies with convex production costs.

Domestic sales γOptimal equity αSOEquil. equity αiTax t
γ = 00.50.50
γ =
γ =
γ =
γ =
γ = 0.50.440.50.25
γ = 0.60.380.50.42
γ =
γ =
γ = 0.900.5433.79
γ = 100.5433.79

Overall, equilibrium equity share αi is larger when firms face convex than linear production costs. Intuitively, firms have stronger incentives to acquire equity on their rivals’ profits when facing convex production costs, since this allows the firm to reduce its total costs. In addition, equilibrium equity αi decreases as the net worth of firm i’s rival, NW, increases.

Welfare analysis. Like in the paper, we next identify the socially optimal output in this context with convex production costs, qSO, and find the corresponding cutoff αSO. The welfare function is given by expression (4), but where firm i’s profit is now πi1(qi+qj)qicqi2. This yields a socially optimal output of


which is positive under all parameter values. As a next step, we evaluate equilibrium output qi in the case of a symmetric equilibrium in the first stage, αi=αj=α, as described above. This yields an equilibrium output


Thus, the condition for which equilibrium output is socially excessive, qi>qSO, is


Cutoff αSO is identical to the cutoff we found with linear production costs and b = 1. Intuitively, the welfare function in expression (4) considers consumer surplus, producer surplus, and environmental damage from production. However, producer surplus collapses to πi+πj2F, which coincides with Vi+αjβNWiCαi+Vj+αiβNWjCαj, where Vi=(1αj)πi+αiπj for firm i. Therefore, convex costs symmetrically affect the firm’s and the social planner’s problem. In contrast, consumer surplus and environmental damage are not affected by the convexity of production costs, ultimately implying that cutoff αSO is unaffected by the type of production costs (linear or convex) that the firm faces. Therefore, the comparative statics of this cutoff remain the same.

Relative to the setting with linear production costs (Table 3 in the main body of the paper), socially optimal equity αSO does not change, while equilibrium equity αi is higher, ultimately yielding lower equity taxes when γ is relatively low, or more stringent taxes when γ is high.

C Extension to joint equity share acquisition

Previous sections considered that every firm independently chooses its equity shares. In some settings, however, firms may negotiate with each other their equity holding. In this appendix, we explore how our findings are affected when firms jointly choose their equity shares in each other’s profits, αi and αj, solving the following joint-maximization problem


Differentiating with respect to αi yields


and similarly after differentiating with respect to αj. As a result, firm i increases its equity αi as much as possible, αi = 0.5, implying that firms acquire weakly more equity when they jointly choose their equity holdings than when they independently do (relative to Table 1 in Section 3.2). Intuitively, under joint equity decisions, every firm’s cost of equity is exactly offset by the equity revenue that its rival receives, ultimately implying that the cost of acquiring equity is nil. This result leads firms to acquire maximal equity on each other, that is, a full merger when firms jointly maximize profits.

Equity taxes. Suppose that the regulator sets a tax t > 0 increasing firms’ cost of equity acquisition. Then, the above problem becomes


Differentiating with respect to αi yields


Every firm i now faces a positive marginal cost from acquiring equity, thus decreasing its equilibrium equity holdings. Since firms acquire maximal equity in equilibrium, our results imply that regulators would set weakly positive taxes under all parameter conditions; as illustrated in Table 8. This table considers the same parameter values as Table 3 showing that, relative to the setting where firms independently acquire equity (Table 3), regulators need to set more severe taxes to induce firms to choose αSO.

Table 8:

Optimal equity taxes under joint profit maximization.

Domestic sales γOptimal equity αSOEquil. equity αiTax t
γ = 00.50.50
γ =
γ =
γ =
γ =
γ = 0.50.440.50.24
γ = 0.60.380.50.14
γ =
γ =
γ = 0.900.590.74
γ = 100.590.74

D Extension to sequential equity acquisition

In this appendix, we consider an alternative equity acquisition game. Here, firms sequentially purchase equity on each others profits. In the first stage, firm i acquires αi on firm j; in the second stage, firm j purchases αj shares on firm i; and in the third stage, based on the profile of equity shares (αi,αj) arising in previous stages, firms compete in Cournot.

We solve this game by backward induction. First, we insert the equilibrium output levels from the last stage to find equilibrium profits, as a function of equity shares αi and αj. Second, we use our condition MBj=MCj as defined in eq. (3) to numerically approximate the follower’s best response function, αj(αi) . Specifically, we consider values of αi in 0.01 increments, within the admissible range of αi[0,0.5], finding the value of αj that maximizes the follower’s profits for every given αi. We then calculate the leader’s profit for each equity level αi, evaluated at the corresponding optimal equity of the follower, αj(αi), identifying which value of αi maximizes the leader’s profits. For comparison purposes, Table 5 evaluates our results using the same parameter values as in the first column of Table 9. In column one in Table 9, we present the leader’s optimal equity share when firms choose their equity sequentially. In column two, we summarize the follower’s optimal equity in this context. Finally, in column three, we show equilibrium equity shares in the simultaneous-move version of the game (as shown in Table 1).

Table 9:

Equilibrium equity share α* when c = 0.3.

Net worth NWLeader, αiFollower, αjSimultaneous game, α*
NW =
NW =
NW =
NW =
NW =
NW =
NW =
NW =
NW =

Table 9 suggests that the leader acquires less equity than in the simultaneous-move game considered in previous sections of the paper, which, in turn, induces the follower to acquire weakly more equity than in the simultaneous game. Intuitively, the leader free rides off the follower’s equity acquisition, thus reducing its equity acquisition. This is because the leader anticipates that the follower will respond by increasing its equity purchases relative to the simultaneous setting of Section 3.2. Additionally, both firms decrease their equity acquisition as the net worth of the company increases, i. e. as we move to lower rows in Table 9.[19]

E Allowing for more firms

In this appendix, we extend our model to a setting with three firms.

Second stage. In this case, firm i solves

maxqi0 Vi=(1αjiαki)πi+αijπj+αikπk

where πi=(abQ)qicqi denotes firm i’s profit and Qqi+qj+qk represents aggregate output and ijk. Intuitively, the first term represent firm i’s share in its own profits, while the second (third) term captures firm i’s share in firm j (firm k , respectively) profits. Differentiating with respect to qi, we obtain best response function


or, more compactly,


where siαij+αik denotes firm i’s equity share in its rivals (firm j and k), and riαji+αki represents the share that all firm i’s rivals hold on this firm’s profits. We find symmetric best response functions for firm j, qj(qi,qk), and for firm k, qk(qi,qj).

Simultaneously solving for qi, qj and qk, yields output


where Aαkj1αji+αij(rk+αik1)αki+αik(ri+rk3), and similar expressions apply for output levels qj and qk. Substituting qi, qj and qk into firm i’s objective function, Vi, gives us


First stage. Anticipating the above profits in the second stage, firm i solves

maxαij,αik Viqi,qj,qk(F+αijβNWj)(F+αikβNWk)+(αji+αki)NWi

Differentiating with respect to αij yields


and differentiating with respect to αik we find


Therefore, firm i faces two first-order conditions, eqs. (6) and (7), and a similar pair of expressions applies to firms j and k, producing a total of six first-order conditions, i. e. generally N(N - 1) expressions where N denotes the number of firms. When firms are symmetric, NWj=NWk=NW for every two firms j and k, expression (6) and (7) coincide and, in addition, all three firms faces the exact same first-order condition, i. e. eq. (6) evaluated at NWj = NW. As a result, all three firms hold the same equity share in equilibrium on their rivals’ profits, that is, αij=αik=α for every firm i and jki/.

We next extend our model to a setting with N firms. For simplicity, we consider symmetric equity shares αi=αj=α for every two firms i and j, as in the symmetric result with three firms presented above. Following the same approach as in the main text, we first identify equilibrium output in this context, then the socially optimal output, and finally compare these two findings.

Equilibrium output. Every firm i solves


where πi[ab(Qi+qi)]qicqi denotes firm i’s profit, Qijiqj represents the aggregate output from all firms other than i, and πj[ab(Qj+qj)]qjcqj denotes firm j’s profit. Differentiating with respect to qi yields best response function


Relative to the best response function identified in Lemma 1 for two firms, qi(Qi) has the same vertical intercept, ac2b, and decreases in its rivals’ appropriation, Q - i; but has a different slope, α(N2)12[α(N1)1]. Since α(N2)12[α(N1)1]N=α22(1α(N1)2>0, the slope is increasing in magnitude in N. Intuitively, competition becomes tougher, and every individual firm reduces its own appropriation more significantly, i. e. firms’ decisions are strategic substitutes to a greater extent. Graphically, the best response function rotates inwards, becoming steeper.

Invoking symmetry in equilibrium, qi=qj=q, we obtain that Qi=(N1)q. Solving for q*, yields the equilibrium appropriation in this N-firm setting


When only two firms compete, N = 2, equilibrium output simplifies to qi(2)=(ac)(1α)(32α)b, which coincides with that in Proposition 1 when equity shares are equal across firms.

We next examine special cases, showing that equilibrium output qi becomes:

  1. Cournot model without equity shares:  qi=ac(N+1)b, i. e. αi = 0 for every firm i.

  2. Cournot model with equally shared equity: qi=ac2Nb, i. e. αi=αj=1N.

Social optimum. The social planner solves


where Qi=1Nqi denotes aggregate output. Note that, since equity shares are symmetric in this setting, they cancel out from the producer surplus, i. e. PS(Q)=i=1NVi=i=1Nπi. Differentiating with respect to qi, we obtain


Invoking symmetry, we find that the socially optimal output level becomes


which collapses to qiSO(2)=(ac)4(b+d)2bγ when only two firms operate, N = 2, thus coinciding with our result in Proposition 3. Socially optimal output qiSO(N) decreases in the number of firms exploiting the resource, N since qiSO(N)N=acN2[b(2γ)+2d]<0. Finally, equilibrium appropriation is socially excessive, qiSO(N)qi(N), if and only if


In the case that only two firms compete in the industry, N = 2, cutoff αSO(2) collapses to αSO(2)1b4d+2b(1γ), thus coinciding with cutoff αSO in Proposition 3.

E.1 Proof of Lemma 1

Firm i solves problem (1), which we can more explicitly write as follows:

maxqi0 (1αj)πi+αiπj

where πi=ab(qi+qj)qicqi and, similarly, πj=ab(qi+qj)qjcqj. Differentiating with respect to qi yields,


Solving for qi, we obtain


Since firms produce weakly positive amounts, we can set the above expression greater or equal to zero, and solve for qj, finding qj(ac)(1αj)b(1+αiαj). Therefore, firm i’s best response function is

qi(qj)=ac2b1+αiαj21αjqj if qj(ac)(1αj)b(1+αiαj)0 otherwise.

E.2 Proof of Proposition 1

From Lemma 1, we found the best response function for firms i and j. Simultaneously solving for qi and qj in qi(qj) and qj(qi), we obtain that the optimal appropriation for every firm i is


This output level is strictly positive since a > c and αi,αj0,12 by definition. We can now differentiate qi with respect to parameters. We can now differentiate qi with respect to parameters. First,


thus indicating that qi is increasing in a. Second,


meaning that qi is increasing in firm j’s equity holding on firm i. Third,


which says that qi is decreasing in its production cost c. Fourth,


which reflects that qi is decreasing in firm i’s equity share on firm j’s profit, αi. Fifth,


which implies that qi is decreasing in the slope of the demand curve, b. Finally, the output difference qiqj is


is weakly positive if and only if αiαj.

E.3 Proof of Proposition 2

We first evaluate equilibrium profits in the second stage of the game, πiqi,qj, by inserting equilibrium appropriation levels found in Proposition 1, qi=(ac)(1αi)b3αiαj and qj=(ac)(1αj)b3αiαj, which yields πiqi,qj=(ac)2(1αi)b3αiαj2. Operating similarly for the equilibrium profit of firm j, we obtain πjqi,qj=(ac)2(1αj)b3αiαj2. Therefore, every firm i solves

maxαi0 (1αj)πiqi,qj+αiπjqi,qj(F+NWjαiβ)+NWiαjβ=(ac)2(1αj)b3αiαj2(F+NWjαiβ)+NWiαjβ

Differentiating with respect to αi yields


with equality if αi>0.

E.4 Proof of Corollary 1

The first term in expression (3) can be interpreted as the marginal benefit that firm i obtains from marginally increasing its equity share in firm j’s profits, αi, that is, MBi2(ac)2(1αj)(3αiαj)3b. This term is increasing in a, decreasing in b and c, and increasing in αj when


which holds as long as αi>2αj. Otherwise, MBi decreases in αj.

We next show that MBi is increasing and convex in αi. Differentiating MBi with respect to αi, we find


Moreover, differentiating MBi with respect to αi again yields


In addition, evaluating MBi at αi = 0, we obtain MBi=2(ac)2(1αj)b(3αj)3, while evaluating MBi at its upper bound, αi=12, yields MBi=2(ac)2(1αj)b52αj3.

Finally, we investigate under which conditions MBiMCi holds for all values of αi. Since MBi is increasing and convex in αi, this is equivalent to the slope of MCi being lower than that of MBi. In particular, this entails MCiαiMBiαi, or NWjβ(β1)αiβ26(1αi)(ac)2b(3αiαj)4. After solving for NWj, yields NWj1β(β1)αiβ26(1αi)(ac)2b(3αiαj)4. Note that when firm i holds no equity on firm j, αi = 0, the above inequality collapses to NWj+, thus holding for all parameter values.

E.5 Proof of Proposition 3

Differentiating with respect to qi in problem (4), we find that qi(qj)=ac2(b+d)bγqj. A symmetric expression applies when differentiating with respect to qj. In a symmetric output profile, we obtain socially optimal output


The numerator of qiSO is positive since a > c by definition. The denominator is positive for all γ<2+2db, which holds for all d, b ≥ 0 since γ0,1 by definition. Therefore, qiSO is positive for all admissible parameter values.

We can now compare equilibrium and socially optimal output, qi and qiSO, by setting qiqiSO=0, and solving for α. We find that qi>qiSO if and only if α < αSO, where cutoff αSO is given by


E.6 Proof of Corollary 2

Setting cutoff αSO > 0 and solving for γ, we find γ<γ112+2db. Setting αSO < 1/2 and solving for γ, we obtain that γ>γ22db. In addition, the difference between cutoff γ1 and γ2 is γ1γ2=12+2db2db=12, implying that γ1>γ2 under all parameter values.

Differentiating cutoff αSO with respect to its parameter values, we obtain



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Published Online: 2019-10-30

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