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 (a−c)2(1−αj)(3−αj)2b≥12βNWj−F 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.

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 π_i is

Equilibrium output – Second stage. Using this definition of π_i in problem (2), and differentiating with respect to q_i, 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, α_i > 0 but α_j = 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 q_i and q_j, 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, π_i, and obtain πi(αi,αj), which represents the profit that firm i earns during the second stage as a function of equity shares α_i and α_j. 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 α_i to obtain firm i’s marginal benefit of acquiring equity, MB_i. This marginal benefit is, however, very large in this setting of convex production costs. For tractability, we do not provide the expression of MB_i 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.

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, q^SO, and find the corresponding cutoff α^SO. The welfare function is given by expression (4), but where firm i’s profit is now πi≡1−(qi+qj)qi−cqi2. 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 α^SO 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+πj−2F, which coincides with Vi+αjβNWi−Cαi+Vj+αiβNWj−Cαj, where Vi=(1−αj)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 α^SO 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 α^SO 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, α_i and α_j, solving the following joint-maximization problem

Differentiating with respect to α_i yields

and similarly after differentiating with respect to α_j. As a result, firm i increases its equity α_i as much as possible, α_i = 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 α_i 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 α^SO.

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 α_i on firm j; in the second stage, firm j purchases α_j 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 α_i and α_j. 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 α_i in 0.01 increments, within the admissible range of αi∈[0,0.5], finding the value of α_j that maximizes the follower’s profits for every given α_i. We then calculate the leader’s profit for each equity level α_i, evaluated at the corresponding optimal equity of the follower, αj(αi), identifying which value of α_i 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 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

where πi=(a−bQ)qi−cqi denotes firm i’s profit and Q≡qi+qj+qk represents aggregate output and i ≠ j ≠ k. 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 q_i, we obtain best response function

or, more compactly,

where si≡αij+αik denotes firm i’s equity share in its rivals (firm j and k), and ri≡αji+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 q_i, q_j and q_k, yields output

where A≡αkj1−αji+αij(rk+αik−1)−αki+αik(ri+rk−3), and similar expressions apply for output levels qj∗ and qk∗. Substituting qi∗, qj∗ and qk∗ into firm i’s objective function, V_i, gives us

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

Differentiating with respect to α_ij yields

and differentiating with respect to α_ik 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 NW_j = 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 j ≠ k ≠ i/.

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≡[a−b(Q−i+qi)]qi−cqi denotes firm i’s profit, Q−i≡∑j≠iqj represents the aggregate output from all firms other than i, and πj≡[a−b(Q−j+qj)]qj−cqj denotes firm j’s profit. Differentiating with respect to q_i yields best response function

Relative to the best response function identified in Lemma 1 for two firms, qi(Q−i) has the same vertical intercept, a−c2b, and decreases in its rivals’ appropriation, Q_- i; but has a different slope, α(N−2)−12[α(N−1)−1]. Since ∂α(N−2)−12[α(N−1)−1]∂N=α22(1−α(N−1)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 Q−i∗=(N−1)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)=(a−c)(1−α)(3−2α)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∗=a−c(N+1)b, i. e. α_i = 0 for every firm i.

2. Cournot model with equally shared equity: qi∗=a−c2Nb, i. e. αi=αj=1N.

Social optimum. The social planner solves

where Q≡∑i=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 q_i, we obtain

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

which collapses to qiSO(2)=(a−c)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=−a−cN2[b(2−γ)+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 α^SO(2) collapses to αSO(2)≡1−b4d+2b(1−γ), thus coinciding with cutoff α^SO in Proposition 3.