Range Best response Maximum Range Best response Maximum
of s1 subsidy Γa2(s1) payoff ¯V2a(s1) of s1 subsidy Γn2(s1) payoff ¯V2n(s1) s1 <sˆ2 β2−4s1 (β2−8s1)2
s1 ≥ˆs2 s1 (β2−2s19)(β2+s1) alls1 (−∞, s1) (β2+s91)2
Policy Switch for Country 2
In Fig. 3.4, the curve namedA1BCA′2 shows the maximized welfare of country 2 given the attracting policy and the curve namedN1BN2 corresponds the non-attracting policy.
country 2’s welfare
s1 ˆ
s2
¯¯ s1 A1
B C
A2
N1
N2
A′1
A′2 0
Fig. 3.4: Country 2’s Payoff in SubgameOC Country 2 chooses the attracting policy only when there holds
V¯2a(s1)>V¯2n(s1). (3-3) In view of the results in Table 3.3, two cases are discussed for solving the above in-equality.
Case 1: When s1 ≥sˆ2, (3-3) can be rewritten as below.
(β2−2s1)(β2+s1)
9 > (β2+s1)2
9 , or 0>(β2+s1)s1.
The above inequality never holds for s1 >sˆ2(>0), so it is better for country 2 to employ the non-attracting policy, i.e., (−∞, s1).
Case 2: When s1 <ˆs2, (3-3) now becomes (β2−s1)2
8 > (β2+s1)2
9 , or s21−34β2s1+β22 >0.
The inequality holds for s1 < (
17−12√ 2)
β2 or s1 > (
17 + 12√ 2)
β2. Since (0 <
)(
17−12√ 2)
β2 < sˆ2 = β52 < (
17 + 12√ 2)
β2, there holds ¯V2a(s1) > V¯2n(s1) for s1 <
(17−12√ 2)
β2. In the following discussion, I define:
¯¯ s1 :=
(
17−12√ 2
)
β2 >0 (3-4)
for brevity of exposition.13 The best-response subsidization policy of country 2 can be summarized as follows.
Γ2(s1)
= Γa2(s1) (= R2(s1)) fors1 <s¯¯1
={Γa2(¯s¯1)} ∪ {Γn2(¯s¯1)}(= {R2(¯s¯1)} ∪(−∞,s¯¯1)) fors1 = ¯s¯1
= Γn2(s1) (= (−∞, s1)) otherwise
The associated reaction curve of country 2 is depicted as the segment R2D and the shaded region excluding the dotted boundary in Figures 3.5 and 3.6. As Krishna (1989) addressing equivalence between quotas and tariffs in duopoly, it is discontinuous ats1 = ¯s¯1.
45◦ sa
1
! s1
s1 s2
R1
R′
1
A1
A2
A3
0 R2
R′
2
D B
! s2
ECC
sCC
1
sCC
2
¯¯ s1
B′
Fig. 3.5: Pure Strategy Equilibrium when β1/β2 ≤βmix in Subgame OC
45◦ sa
1
! s1
s1
s2 R
1
R′
1
A1
A2
A3
0 R2
R′
2
D
B
! s2
ECC
sCC
1
sCC
2
¯¯ s1
F
B′
Fig. 3.6: Mixed Strategy Equilibrium when β1/β2 > βmix in Subgame OC
Comparison of the two figures indicates that the pure-strategy equilibrium is possible if and only if ¯¯s1 ≥sCC1 holds, i.e., β1/β2 ≤βmix(
:= 64−45√
2∈(0,1))
holds 14. As shown in Krishna (1989), it is straightforward to prove the following proposition.
Proposition 3.1. Depending on the value of β1/β2, there emerge two types of equilibria for subgame OC as follows.
P3.1.1 Forβ1/β2 ≤βmix(
:= 64−45√
2∈(0,1))
, the same pure-strategy equilibrium as in subgame CC is realized.
P3.1.2 Otherwise, a mixed-strategy equilibrium is realized where country 1 (having em-ployed O) chooses s¯¯1 with probability unity and country 2 (having employed C) randomizes over R2B(¯s¯1) and (−∞,¯¯s1).
The above Proposition implies that when the firm in the capital liberalizing country is much less productive than the rival firm, the capital liberalizing country behaves as in subgameCC, subsidizing the domestic firm to maximize its own welfare. The rival country, which is a much more productive country has strong incentive to subsidize the national firm itself. Otherwise, the capital liberalizing country sets a lower subsidy than the welfare-maximized subsidy in subgame CC, since the rival country is not so much productive and may set a lower subsidy or tax to let the firm move abroad to achieve higher welfare.
14It is straightforward to derivesCC1 −¯¯s1= 4β152 {(β1
β2
)−(
64−45√ 2)}
. One should also noteβmix>
1/3, as shown byβmix−13 = (64−45√
2)−13 ∝ 191−135√
2∝ 191135−√ 2>0.
3.4.4 Characterization of Mixed-Strategy Equilibria for Subgame OC
Since the pure-strategy equilibrium for subgame OC is the same as in subgame CC, I focus on the mixed-strategy equilibrium as β1/β2 > βmix shown in Figure 3.6 and characterize its comparison with subgame CC.
Mixed-Strategy Equilibria
Start at the equilibrium subsidy pair given by point ECC with sCC2 > sCC1 .15 Given sCC1 , country 2 finds it better to let firm 2 go abroad and thus lowers its subsidy rate belowsCC1 . When country 2 chooses the subsidy rate slightly below sCC1 shown by pointF, country 1 also has an incentive to cut its subsidy rate so as to spare the subsidy expenses to firm 2, which is then matched by the further subsidy decrease by country 2. This race of subsidy reduction continues along the 45◦ line until reaching point B where country 2 finds it indifferent to keeping firm 2 at home with R2B(¯s¯1) (point D) and letting it move abroad with s2 ∈(−∞,s¯¯1) (the segment BB′ ) when country 1 chooses ¯¯s1.
Let me further characterize this mixed strategy of country 2. For this purpose, let ρ represent the equilibrium probability of country 2 choosing R2B(¯¯s1) and 1−ρ the prob-ability of its choosing other subsidy rates smaller than ¯s¯1. The equilibrium value of ρ can be obtained by analyzing country 1’s optimization behavior. DenoteW1OCm(s1) as the expected welfare of country 1 in the mixed-strategy, where the superscript Cm represents that country 2 employs a mixed strategy on export subsidies. Given ρ,W1OCm(s1) is given by
W1OCm(s1) =ρW1(
s1, R2B(¯s¯1))
+ (1−ρ)V1a(s1)
= ρ 9 ·(
β1+ 2s1−R2B(¯s¯1)) (
β1−s1−R2B(¯s¯1)) + (1−ρ)
((β1+s1)2
9 −s1(β1+β2+ 2s1) 3
) ,
where use was made of (2-26) and (3-1). Differentiation with respect to s1 yields 9dW1OCm(s1)
ds1 =ρ(
β1−4s1−R2B(¯¯s1))
+ (1−ρ) (−β1−3β2−10s1). Since it must hold that lims1→¯¯s1
dW1OCm(s1)
ds1 = 0, ρ can be derived as ρ = 4 (β1 + 3β2+ 10¯s¯1)
8β1+ 11β2+ 25¯s¯1 = β1/β2+ 173−120√ 2 2β1/β2+ 109−75√
2, (3-5)
15The following discussion assumessCC2 > sCC1 . However, the same analysis applies even when sCC2 ≤ sCC1 with some modifications in Figure 3.6.
by virtue of ¯¯s1 = (17 −12√
2)β2 in (3-4). Using ρ in the above equation, the expected welfare of each country in the mixed-strategy equilibrium is given by
W1OCm :=ρW1(
¯¯
s1, R2B(¯s¯1))
+ (1−ρ)V1a(¯s¯1), (3-6) W2OCm :=W2(
¯¯
s1, R2B(¯s¯1))
= (β2+ ¯s¯1)2
9 . (3-7)
3.4.5 Welfare Comparison between Subgames CC and OC
As implied by the discussion in the previous section, country 2 banning inward direct investment is sure to get better off in subgame OC than in subgame CC, i.e., W2OCm >
W2CC. This is because its welfare at pointD is strictly higher than atE along its reaction curve R2R2′ in Figure 3.6.
However, as to country 1, it is not clear at the first glance at the figure whether it is better off in subgame OC than in subgame CC. This is because, compared with E, there are (i) the losses both from the higher subsidy of country 2 and the failure to optimize its subsidy rate when country 2 sets R2(¯s¯1), and (ii) the gain from lowering the subsidy and the loss from rent outflow due to subsidizing firm 2 when country 2 chooses s2 ∈(¯¯s1,−∞).
However, some tedious calculations in Appendix 3.A show that country 1 gets worse off in subgame OC than in subgame CC.
Lemma 3.3. Givenβ1/β2 > βmix (i.e., with a mixed-strategy equilibrium), W1CC > W1OCm holds.
Therefore, in the mixed-strategy equilibrium, unilateral capital liberalization makes the country liberalizing capital worse off and the country banning capital inflow better off.
Note that subgame CO can be solved in the same way as subgame OC. That is, sub-game CO yields pure-strategy equilibrium for β1/β2 ≥1/βmix and mixed-strategy equilib-rium for β1/β2 < 1/βmix. The other correspondent results also apply. Lastly, I turn to examine the equilibrium in subgame OO.
Subgame OO is an extension of the BS model explored by Janeba (1998) in which both exporting countries liberalize capital, i.e., the two exporting firms can freely choose their locations for production. The analysis makes sense only when both countries have already decided to accept inward direct investment from abroad.
Using the same approach as the previous section, I reexamine the result in Janeba (1998). When both exporting countries have liberalized capital, each firm’s strategic loca-tion choice depends on the subsidy rates chosen by the two countries. The country offering a higher subsidy can attract both firms, but suffers from the foreign rent outflow. Taxation can restrain this rent outflow, but induces both firms to go abroad, leading to a loss in tax revenue. To derive country 1’s best response subsidy, I first show the associated welfare as below.
V1(s1, s2) =
W1(s1, s1)−s1x2(s1, s1) whens1 > s2 W1(s1, s2) whens1 =s2 W1(s2, s2) +s2x1(s2, s2) whens1 < s2
Based on the analyses in the previous section, country 1’s best-response subsidy in subgame OO is identical to its best attracting policy in subgame OC when s1 > s2, and its best non-attracting policy in subgame CO when s1 < s2. Thus, country 1’s best response subsidy againsts2 and the associated maximized welfare level are summarized in the following table.