4.3.1 Second-Stage Equilibrium
As mentioned earlier, the equilibrium is solved by backward induction from the second-stage. Firm i’s reaction function denoted byriC(xj, si) is equivalent to the one in the BS model. The superscriptCshows the variables associated with the equilibrium values under the cross-country shareholding structure:
riC(xj, si) = 1
2(a−ci+si−xj) = riB(xj, si) (i, j = 1,2; j ̸=i). (2-19) Denote x∗iC(si, sj) (i, j = 1,2; j ̸=i) as firm i’s equilibrium output under cross-country shareholding that is expressed by
x∗iC(s) = 1
3(βi+ 2si−sj) =x∗iB(s) (i, j = 1,2; j ̸=i). (2-20) The equilibrium profit of firm i can be rewritten as πi∗C(s) = πi(x∗iC(s), x∗jC(s), si) = πi∗B(s).
Under the cross-country shareholding structure, the familiar comparative statics re-sults are equivalent to the BS model. This is because no changes in the bilateral mutual shareholding structure affect either firm’s output decision.
4.3.2 Government’s Subsidy Incentive
In the first stage, the governments can predict the resulting second-stage equilibrium, given their own choices regarding the subsidies. Countryi’s welfare is now given by
Wi∗C(s,σ)def= σiπi∗C(s) + (1−σj)πj∗C(s)−six∗iC(s).
Each government maximizes the national welfare by choosing the optimal export sub-sidy, taking into account the responses of both firms.3 The Nash solution for the FOC should satisfy
0 = ∂Wi∗C(s,σ)
∂si
=σi∂π∗iC
∂si + (1−σj)∂πj∗C
∂si −xi−si∂x∗iC
∂si
=σi (
xiP′(X)∂x∗jC
∂si
+xi )
−xi−si∂x∗iC
∂si
+ (1−σj)xjP′(X)∂x∗iC
∂si
, (4-3)
3The SOC for each country’s welfare maximization can be examined by using (4-5):
∂2Wi∗C(s,σ)
∂s2i = 1
9[8σi−10−2σj]<0,
which shows that the welfare function of each country is strictly concave with respect to its own export subsidy.
where use was made of (2-10) and (2-11). The terms on the right-hand side of (4-3) show the decomposition of strategic export subsidization under the cross-country shareholding structure. The subsidy incentive is easy to understand when it is compared to the standard form in the BS model (σi = 1 fori= 1,2). For this purpose, (4-3) is rewritten as below.
∂Wi∗C(s,σ)
∂si
=xiP′(X)∂x∗jC
∂si −si∂x∗iC
∂si
−(1−σi)xiP′(X)∂x∗jC
∂si −(1−σi)xi+ (1−σj)xjP′(X)∂x∗iC
∂si .
The terms on the first line show the subsidy incentives found in the BS model in (2-12), while those on the second line show the new sources of subsidy incentives specific to cross-country shareholding. For later reference, the terms on the second line are called the additional subsidization incentivesunder cross-country shareholding and are denoted as IiC(s,σ) for country i, that is,
IiC(s,σ) = −(1−σi)xiP′(X)∂x∗jC
∂si −(1−σi)xi+ (1−σj)xjP′(X)∂x∗iC
∂si . (4-4) The above three terms on the right-hand side are all strictly negative in view of (2-8) and (2-9). The first term −(1−σi)xiP′(X)∂x∗
C j
∂si represents the cross rent-shifting effect. Export subsidy to the domestic firm, through the standard rent-shifting effect, increases its profit, but it leads to an increase in the dividend given to the rival firm. This effect becomes smaller in absolute terms as the domestic firm’s own shares increases. In other words, the higher the domestic firm’s own share, the less the government cares about the outflow of the domestic firm’s rent due to the cross rent-shifting effect.
The second term−(1−σi)xi shows thesubsidy outflow effect, for it represents the portion of the subsidy expense going abroad as increased dividend to the foreign residents.
Further, its absolute value decreases as the domestic firm’s own share increases.
The last term (1−σj)xjP′(X)∂x∂s∗iC
i shows the dividend suppression effect, for it represents the portion of the decrease in the dividend received from the shared rival firm.
The above three effects weaken the subsidy incentives in the presence of cross-country shareholding.4
From (2-21) in the linear demand function, (4-3) can be written as 0 = ∂Wi∗C(s,σ)
∂si
= 1
3[(4σi−3)xi−2si−2(1−σj)xj]. (4-5) Let RiC(sj,σ) represent country i’s reaction function. Since no changes in the cross-country shareholding structure affect the comparative statics results for the market out-comes, it is straightforward to derive the effect of a change in the cross-country shareholding
4Welzel (1995) and Dick (1993) also identified three similar effects on the optimal export subsidization.
Here, these three effects are used to conduct a comparison with the cross-firm shareholding case for later analysis.
structure on the optimal response subsidy. In view of the SOC for national welfare maxi-mization, an application of the implicit function theorem using the result in (4-5) yields
∂RiC(sj,σ)
∂σi =−∂2Wi∗C/∂σii∂si
∂2Wi∗C/∂s2i ∝ ∂2Wi∗C
∂σii∂si = 4
3xi >0.
An increase in the domestic residents’ share over the domestic firm makes the government care more about the profit of the domestic firm, and thereby, strengthens the government’s strategic export subsidy incentive.
Similarly, an increase in the foreign ownership of foreign firm leads to
∂RiC(sj,σ)
∂σj =−∂2Wi∗C/∂σjj∂si
∂2Wi∗C/∂s2i ∝ ∂2Wi∗C
∂σj∂si = 2 3xj >0
which implies that when there is a decrease in the domestic residents’ claim over the foreign firm’s profits, the domestic country’s government then cares less about the decrease in the foreign firm’s profits, which gives rise to a stronger export subsidy incentive a la Brander-Spencer.
I summarize the above results into the following Lemma.
Lemma 4.1. An increase in the residents’ ownership share in the domestic firm induces both governments to increase the strategic export subsidy rates.
The full-game Nash equilibrium subsidy is a solution to the following reaction function.
RiC(sj,σ) = 4σj −4σi−1
10−8σi+ 2σj sj +(4σi−3)βi−2(1−σj)βj 10−8σi+ 2σj , where βi =a−2ci+cj(i, j = 1,2;j ̸=i) is defined in Section 2.5.
The equilibrium subsidy, denoted assCi (σi, σj) depends on the cross-country sharehold-ing structure.
sCi (σ) = 4(10σi+ 4σj −6σiσj −7)βi+ (8σi+ 20σj−12σiσj −17)βj
3(33−20σ1−20σ2+ 12σ1σ2) . (4-6) Proposition 4.1. The equilibrium export subsidy rate under the cross-country shareholding structure is strictly lower than that in its absence, that is, sCi (σ)< sCi (1) =sBi .
Proof: See Appendix 4.A. 2
To clarify how the cross-country shareholding structure governs the equilibrium export subsidy profile, let me focus on the marginal subsidy rate whensi = 0 and investigate what factors induce the government to impose export subsidies. In view of (4-5), it follows that
∂Wi∗C(s,σ)
∂si
si=0
= 4σixi+ 2σjxj−3xi−2xj. (4-7)
Denote the market share of firm i asθi =xi/X (i= 1,2) where θ1+θ2 = 1. Thus the government has an incentive to subsidize the domestic firm if and only if 4θiσi+2θjσj−(3− θj) > 0. (4-7) also yields the following lemma specific to the cross-country shareholding structure.
Lemma 4.2. Under the cross-country shareholding structure, the government’s subsidy incentive is dependent on market share θi.
σ2
σ1 (12,12)
1
1 S1
S1′ S2
S2′ (1,1) sC1 >0
sC1 <0
sC2 >0
sC2 <0
5 6
5 6
Fig. 4.1: Government’s Subsidy Incentive under Cross-Country Shareholding Figure 4.1 that allows the pairs of the shares letting both countries to find a zero subsidy optimal with respect to (4-7), where curve S1S1′ stands for country 1 and curve S2S2′ for country 2 in the case of symmetric cost conditions. The unit square[1
2,1]
×[1
2,1] is divided into four areas, each representing both governments’ incentives for subsidization or taxation. Figure 4.1 implies that both governments have an incentive to provide a subsidy only when the domestic residents’ share of the domestic firm is sufficiently high as represented by the shaded area. The BS model corresponds to the top corner (1,1). In the presence of cross-country shareholding, each government may rather tax the domestic firm even though the residents’ equity share constitutes over half of the total share. In fact, when the residents’ share is half for both firms, the governments would definitely tax the exports. This is exactly the case when there is a single exporting country holding two firms. When the two domestic firms compete in the export market, the competition becomes excessive from the view point of joint profit maximization. The government thus has an incentive to suppress, rather than promote the competition so as to maximize national welfare.