Discussion Paper No. 693

PRICE UNDERTAKINGS, VERs, AND
FOREIGN DIRECT INVESTMENT
THE CASE OF FOREIGN RIVALRY
Jota Ishikawa
and
Kaz Miyagiwa

July 2007

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Price undertakings, VERs, and foreign direct investment
The case of foreign rivalry#

Jota Ishikawa* and Kaz Miyagiwa**

Abstract
Antidumping (AD) petitions are often withdrawn in favor of VERs and price undertakings. We
compare foreign firms’ incentive to engage in foreign direct investment (FDI) under a VER anda price undertaking, with special emphasis on foreign rivalry. We show that a VER is less likely
to induce FDI than a price undertaking or AD. Thus, the importing country can increase thelevel of protection by replacing an AD duty with a VER. This may account for the GATT ban
on VERs, given the proliferation of AD cases during the 1990s.
Keywords: FDI, price undertakings, antidumping, VER, WTO
JEL Classification code: F1

Corresponding Author: Kaz Miyagiwa, Institute of Social and Economic Research, Osaka
University, Mihogaoka, Ibaraki, Osaka 567-0047, Japan. E-mail: kmiyagiwa@gmail.com#

We are grateful to an anonymous referee and Ig Horstmann, Co-Editor, for their helpful suggestions that led to
substantial improvements. We also thank Richard Baldwin, Yongmin Chen, and seminar participants at the
University of Colorado, the Graduate Institute of International Studies, and the City University of Hong Kong fortheir comments. We acknowledge financial support from the Ministry of Education, Culture, Sports, Science andTechnology of Japan under the 21st Century Center of Excellence Project. Ishikawa also wishes to thank the
Nomura Foundation and the Japan Securities Scholarship Foundation for financial support. We alone areresponsible for errors.
*
Faculty of Economics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan; jota@econ.hit-u.ac.jp**
ISER, Osaka University, Osaka, Japan, and Department of Economics, Emory University, Atlanta, GA, U.S.A.

1. Introduction
Over the last-quarter century, use of antidumping (AD) has spread from a handful ofusers (primarily the United States and the European Union) to more than 70 countries (Prusa,
2001). In reality, however, AD petitioners often withdraw their petitions in favor of direct
settlements with exporting firms or governments. For example, during the 1980-85 period over
one-third of U.S. AD petitions were withdrawn in favor of VERs or voluntary export restraints(Prusa, 1992). During the same 1980-1985 period nearly half the AD petitions filed in the EU
were settled through price undertakings, which are voluntary price increases offered by foreign
firms to offset the injury to domestic producers due to alleged dumping (Messerlin, 1989).
More recently, the U.S. and Canada have started to settle AD cases through priceundertakings.1
In this paper we examine the relative effect of a price undertaking and a VER on
foreign firms’ incentive to engage in FDI, with special emphasis on foreign rivalry. There is awell-developed literature on “protection-jumping” FDI, but most of this literature assumes one
foreign firm competing with one home firm.2 Such a framework cannot reveal any influence
foreign rivalry may have on incentives to engage in protection-jumping FDI.
To capture the role of foreign rivalry in firms’ choice between exporting and FDI, we
consider a model in which two firms from the same foreign country compete with one homefirm in the home country. Our analysis offers two main results. A first is that a price1

Use of VERs was prohibited under the GATT in 1994 (Agreement on Safeguards, Article 11) and the existing
VERs have been phased out. In contrast, price undertakings remain in full compliance under GATT and WTO rules(Article 8). See Moore (2005) for more institutional backgrounds.
2
See, e.g., Horstmann and Markusen (1987), Motta (1992), Ishikawa (1998), Blonigen, Tomlin and Wilson. (2004),
and Belderbos, Vandenbussche, and Veugelers (2004). Miyagiwa and Ohno (1998) and Blonigen and Ohno (1998)
study aspects of foreign rivalry in other contexts.

2
undertaking leads to more FDI than a VER. To understand this result intuitively, suppose that
one foreign firm, under a price undertaking, switches from exporting to FDI while the otherexports. Freed from the constraint, the investing firm expands output, depressing the price forthe exporter. However, since the export price is not allowed to fall under a price undertaking,the exporter must curtail more exports to keep the price up. Thus, one firm’s decision to engage
in FDI generates negative externalities on the exporting firm, giving each firm an incentive toinvest before the rival does.
With a VER, one firm’s choosing FDI over exporting allows the foreign government toreallocate the export quota the firm no longer needs to the exporting firm, thereby relaxing theconstraint confronting the latter. Thus, FDI by one firm generates positive externalities on theexporter, thereby diminishing each firm’s incentive to invest. This difference establishes our
first result.
Our second result concerns importing country welfare. We consider a case in which thehome government settles an AD case through a VER or a price undertaking under the politicalconstraint that the home firm should not be made worse off as a result of the settlement. This
constraint seems realistic, for otherwise the home firm would never withdraw its AD petition infavor of such a settlement. In such a setting we show that home country welfare is never lower
with a VER – and can be higher – than with a price undertaking. The explanation comes fromour first result. Since a VER is less likely to induce FDI, which harms the home firm, than aprice undertaking, the home country government faces a less stringent policy constraint under a
VER than under a price undertaking. Optimization under a looser constraint gives rise to
greater welfare.

3
An interesting policy implication is that the home country government can negotiate amore protectionist outcome when it settles AD cases for VERs. Thus, the 1994 GATT decision
to prohibit use of VERs can be interpreted as a policy to curb the spread of protectionism.
There is already a large literature on VERs. Price undertakings have been receiving
increasing attention from trade economists; see, for example, Vandenbussche and Wauthy(2001) and Pauwels and Springael (2002). Our study is particularly related to that of Belderbos,
Vandenbussche, and Veugelers (2004). These authors consider a three-stage game, in which
AD authorities first choose between an AD duty and a price undertaking, the foreign firmchooses between exporting and FDI, and then the foreign and the home firm compete in prices
in the home market. FDI reduces the foreign firm’s production cost but may not occur due tohigh setup cost. In such cases, welfare-maximizing AD authorities choose an AD duty over a
price undertaking to induce FDI, but if concerned with protection of the home firm, theychoose a price undertaking to prevent FDI. Thus, there the main focus is on the government’s
choice between an AD duty and a price undertaking, with one foreign firm, while here it is on acomparative effect of a price undertaking and a VER in the presence of foreign rivalry.
The differences between a price undertaking and a VER are explored in Moore (2005),
although in the absence of FDI. Unlike here he assumes differentiated-goods Bertrand duopoly
and shows that a VER leads to a more collusive outcome than a price undertaking.
The reminder of the paper is organized in 6 sections. The next section outlines themodel and demonstrates the equivalence among the three policy tools in the absence of FDI.
Sections 3, 4, and 5 examine AD, price undertakings, and VERs, respectively. Section 6
discusses policy implications. Section 7 concludes.

4

2. Environment and basic results
To capture foreign rivalry, the focus of our analysis, we consider a model in which two
foreign firms from the same country compete with one home firm in the home market.
Additional assumptions simplify the analysis. Firstly, unlike in Belderbos et al. (2004) all firms
are symmetric and have zero production costs regardless of the location choices the foreign
firms make. Thus, FDI offers no production cost advantages to foreign firms. However, FDI
requires a fixed setup cost k > 0. Thus, FDI is never chosen over exporting under free trade.
Secondly, unlike in Moore (2005) the firms produce homogeneous goods and play a Cournot
game. Thirdly, home market demand is linear and written
p = a – (x1 + x2 + xh),
where a > 0 and xh denotes the home firm’s output and x1 and x2 the two foreign firms’
outputs.
Cournot competition has an advantage over Bertrand competition in that it establishes
the equivalence among AD, the VER and the price undertaking, which makes it possible toisolate the effect of foreign rivalry in policy comparisons. This three-way equivalence is easy toshow using Figure 1 in case of only one foreign firm.
<Please place Figure 1 around here.>

Letting xh and xf denote the home and the foreign firm’s output, respectively, suppose that theCournot-Nash equilibrium under free trade is indicated by point N, where the home firm’s bestresponse function rh intersects that of the foreign firm rf. The imposition of AD shifts the5foreign firm’s best-response function to rft, and the equilibrium to Nt. Converting this AD duty
to a VER set equal to the volume of imports under the AD duty has no effect on the homefirm’s best-response function. However, the foreign firm’s best-response function consists ofthe vertical dashed line AB and the segment of rf. above point A. The equilibrium remains at
Nt, however, proving the equivalence between the two instruments.3
Alternatively, convert the AD case to a price undertaking with the price p that prevails-

at Nt. This conversion again has no effect on the home firm’s best-response function, but theforeign firm’s best-response function now comprises the thick line through Nt, along which theprice is constant at p, and the segment of rf below point C. The Nash equilibrium remains at-point Nt, however, thereby establishing the equivalence between AD and a price undertaking.
This result extends to the case of two foreign firms as we show in Appendix A. Thus,

Proposition 1: In the absence of FDI, there exist a VER and a price undertaking, which areequivalent to a given AD duty.4

In Figure 1, if the foreign firm were allowed to engage in FDI, it would choose to doso, in which case the equilibrium would be at point N as under free trade. Therefore, the foreign
firm’s incentive to engage in FDI must be the same, whether the firm faced a VER or a price3

Hwang and Mai (1988) were the first to show the equivalence between the tariff and the VER. See also Ishikawa(1994).
4
With Bertrand competition a VER and a price undertaking are not equivalent, as shown by Moore (2005). Krishna(1989) showed the non-equivalence between a tariff (or an AD duty) and a VER.

6
undertaking. This conclusion however does not extend when there are more than one foreign
firm as we demonstrate in the next three sections.

3. Antidumping
We consider a two-stage game, in which two foreign firms simultaneously choose
between exporting and FDI, and then play a quantity-setting game with a home firm in thehome country.5 Since profits depend on the location choice the foreign firms make in the firststage, we first examine all possible second-stage games, and then the foreign firms’ FDI
decisions.
If both firms export under an AD duty t, each foreign firm produces (a – 2t)/4 and thehome firm (a + 2t)/4 units at the price(1)p(t) = (a + 2t)/4.

The profit to each foreign firm is(2)π(t) = (a – 2t)2/16,

and that to the home firm is(3)

Π(t) = (a + 2t)2/16.

If both foreign firms invest, the AD duty no longer applies, so the equilibrium profits are a2/16
for the home firm and a2/16 – k for the foreign firms as can easily be checked. If only one
foreign firm invests, the investor earns (a + t)2/16 – k, while the exporter earns (a - 3t)2/16. The

5

Following convention, we ignore what might be going on in the foreign country market. It is straightforward to
introduce the segmented foreign market, where the price is higher than in the home market, so as to motivate why
there is an AD duty, to begin with.

7
following table presents the foreign firms’ profits, where I denotes investing, E denotes
exporting, and the first entries are the profits to the row player, the convention we retain
throughout:

AD Duty (t)IEIa2/16 – k, a2/16 – k(a + t)2/16 – k, (a – 3t)2/16

E(a – 3t)2/16, (a + t)2/16 – k(a – 2t)2/16, (a – 2t)2/16.

In the first stage of the game the foreign firms simultaneously choose between I and E,
anticipating the profits summarized above. We impose the following conditions on parametervalues:(i) Simultaneous investments are profitable, or a2/16– k > 0,(ii) AD duties are never prohibitive, or t < a/3
Figure 2 presents the equilibrium outcomes for relevant values of k and t (see Lemma 1
in Appendix B for details). The figure shows, not surprisingly, that the foreign firms choose
FDI over exporting when the cost of investment is relatively low and the AD duty is relatively
high.
<Insert Figure 2 around here>

4. Price undertakings

8
Suppose that (E, E) is the equilibrium outcome under AD. Using (1) we can relate theprice undertaking to the AD duty byp = (a + 2t)/4,-

which, given that t < a/3, ranges over the interval (a/4, 5a/12). It is convenient to consider twocases separately.

Case I: a/4 < p < a/3.

-

This case corresponds to t ∈ (0, a/6). If both foreign firms invest, each earns a2/16 – k as in AD
cases. If both export, each earns !(p) = p(a - 2p)/2 (see Appendix A). With only one firm

-

-

-

investing, the exporter takes the home firm’s and the investor’s output, xi to choose output xe
under the constraint that the price not fall below p:-

p = a – xh – xi – xe ≥ p.

-

Calculations show that in the symmetric Nash equilibrium
xe = a – 3p < p = xi = xh.

- -The investor earns!i(p) = p2 – k,--while the exporter earns!e(p) = p(a – 3p).

-

-

-

The table below summarizes the profits to the foreign firms in all second-stage games.

9
Price undertaking (Case I): a/4 < p < a/3

-

I

E

I

a2/16 – k, a2/16 – k

p2 – k, (a – 3p)pE(a – 3p)p, p2 – kp(a – 2p)/2, p (a – 2p)/2

-

--

-

-- -

-

-

-

Case II: a/3 < p < 5a/12-

This case occurs when an AD duty was set higher, i.e., a/6 < t < a/3. In this case, p is so high-that, when one foreign firm chooses FDI, the equilibrium price falls below p even if the other-foreign firm has no sale. That is, one firm’s decision to engage in FDI forecloses the homemarket foreclosed to exports. Because of foreclosure there is even a stronger incentive to be afirst to invest than in Case I. This yields the following table.

Price undertaking (Case II): a/3 < p < 5a/12

-

I

E

I

a2/16 – k, a2/16 – k

a2/9 – k, 0

E0, a2/9 – kp(a – 2p)/2, p (a – 2p)/2

-

-

-

-

Figure 3 depicts the equilibrium outcomes for both cases (see Appendix B for details).

10<Insert Figure 3 around here.>
A comparison with Figure 2 shows that for t ≤ a/6 the equilibrium (E, E) occurs in a larger setof parameter values. Intuitively, the choice between exporting and FDI depends on two effects.
A first is standard; since there are no duties to pay the benefits from FDI are smaller under aprice undertaking than under AD. A second is due to foreign rivalry. Under AD one foreign
firm’s decision to engage in FDI decreases the equilibrium price, while under a priceundertaking the exporting firm must curtail more exports to prevent the price from falling. This
externality makes FDI more attractive under the price undertaking. A comparison shows thatfor t ≤ a/6 the externality effect is less important.
For t > a/6, the externality effect is dominant because of home market foreclosure toexports. Thus, (I, I) is an equilibrium at all relevant parameter values. However, when the costof investment is sufficiently high (E, E) can also be an equilibrium because no firm wants toinvest if the rival does not, implying multiple equilibria as shown in Figure 3. We summarize
the result in

Proposition 2: For t < a/6, a price undertaking induces both firms to export in a broader set ofparameters than the equivalent AD duty. For t > a/6, a price undertaking can induce more FDI
than the equivalent AD duty.

5. VERs
The VER is related to the AD duty by the equivalence formula:v = (a – 2t)/2,

11which ranges over v ∈ (a/6, a/2) given t < a/3 (see Appendix A). Under the VER agreement,
total exports from the foreign country cannot exceed v; that is, x1 + x2 ≤ v. Assume that theforeign government allocates the export quota equally to the two firms when both export. If one
firm invests, it no longer needs its export quota so the foreign government can reallocate it tothe other firm. As with a price undertaking, consider two cases.

Case I: v ∈ (a/3, a/2).
This case corresponds to t ∈ (0, a/6). If both foreign firms export, each firm faces the quota v/2.
Given the equivalence, the equilibrium outcome is the same as under the price undertaking. If
the two firms invest instead, each foreign firm earns a2/16 – k. If only one firm invests, theentire quota, v, is given the sole exporter. In the present case, since v > a/3 the total quota
exceeds the per-firm symmetric Cournot equilibrium output, a/4, implying that the VER is notbinding for the exporting firm. With all firms playing a Cournot game unconstrained, theequilibrium profits to the investor and the exporter, respectively, are:!i = a2/16 – k, and !e = a2/16.
We have obtained the following table.
VER (Case I): v ∈ (a/3, a/2)
IE

I

a2/16 – k, a2/16 – k

a2/16 – k, a2/16

E

a2/16, a2/16 – kv(a - v)/2, v(a - v)/2

12
We show in Appendix B that (E, E) is the only equilibrium of this game (see Lemma 4).
Turn next to Case II, in which v ∈ (a/6, a/3]. Notice that for v ∈ [a/4, a/3]
corresponding to t ∈ [a/6, a/4], the VER is still not binding for the exporter when the other firmchooses FDI. Therefore, (E, E) is the unique equilibrium. For v ∈ (a/6, a/4) corresponding to p-

∈ (3a/8, 5a/12) or t ∈(a/4, a/3), however, when one foreign firm invests, the VER becomesbinding for the exporter. In equilibrium the exporter produces v units of output while the homefirm and the investing firm produce
xh = xi = (a – v)/3.
The exporter earns !e(v) = v(a – v)/3, and the investor makes !i(v) = (a – v)2/9 - k. Given therange of v, we have !i(v) > !e(v), so there is now an incentive to invest if the other does not(provided that k is small). The payoffs from all the second-stage games are shown in the tablebelow.
VER (Case II): v ∈ (a/6, a/4)
IE

I

a2/16 – k, a2/16 – k(a – v)2/9 – k, v(a – v)/3

E

v(a – v)/3, (a – v)2/9 – kv(a – v)/2, v(a – v)/2

<Please insert Figure 4 around here>
Figure 4 depicts the equilibrium with VERs (see Lemmas 4 and 5 in Appendix B for details). A
comparison of figures 2 – 4 shows that settling an AD case for a VER leads to less FDI than fora price undertaking.

13

Proposition 3. The set of parameter values in which both firms choose exporting over FDI
under VERs is a strict superset of those under AD and price undertakings.

The proposition says that the home country government can, for given k, set a higher level ofprotection with a VER than with AD or a price undertaking without triggering FDI, implyingthat settling an AD case for a VER can lead to a hidden increase in protection.

6. Policy implications
In this section we discuss policy implications of our model. In particular, we consider
the following scenario. The home firm petitions for AD protection, but is ready to withdraw thepetition if a price undertaking (or VER) agreement can be worked out with the foreign firms or
the foreign government. Suppose that with AD the equilibrium outcome is (E, E). Now, ourquestion can be stated. Should the home government choose a price undertaking or a VER, if it
wants to settle the AD case under the condition that the home firm is not to be hurt by thesettlement? We show that the home country is generally better off with a VER than with a priceundertaking.
Suppose that we are in Case I. If the government is constrained to stay within Case I, a
price undertaking and a VER yield the same welfare. To see this, observe that (E, E) is theequilibrium outcome under the equivalent VER whenever it is under AD. Further, in Case I atighter VER decreases welfare (see Appendix C) while a looser VER harms the home firm.
Thus, the government has no choice but to choose the equivalent VER, given the constraint that14
it stays in Case 1. Similarly, when settling an AD case for a price undertaking, the best the
government can do is to set the equivalent price target. Thus, the price undertaking and the
VER yield the same home country welfare.
Suppose next that the government can choose a protection level beyond the parameter
ranges of Case I. By Proposition 3, the set of parameter values resulting in (E, E) under a VER
is a strict superset of that under a price undertaking. Operating under a larger constraint set the
government can do better with a VER than with a price undertaking. Thus,

Proposition 4. A VER (weakly) welfare-dominates a price undertaking.

7. Concluding remarks
In this paper we study the foreign firms’ incentives to engage in FDI under a VER anda price undertaking. While there is a well-developed literature on protection-jumping FDI, ourapproach is novel in that it focuses on foreign rivalry as a determinant of FDI. Our main result
is that a price undertaking gives a greater incentive to engage in FDI than a VER. The reason isthat with a price undertaking one firm’s decision to engage in FDI depresses the export price,
forcing the exporting firm to contract exports, whereas with a VER it relaxes the constraint
facing the exporting firm as the export quota no longer needed by an investing firm can bereallocated to the exporting firm.
We also explore policy implications of our analysis. Since a VER is less likely to
induce FDI than AD and a price undertaking, the importing country government can negotiate
a more protectionist outcome without triggering FDI by settling an AD case for a VER

15
agreement. Thus, our analysis implies that the GATT ban on VERs was a sensible decision tostop the spread of increases in protection.
But if GATT/WTO decisions are nothing but a reflection of member country
preferences, one wonders why members agreed to the ban in the first place. To address thisquestion, consider an extension of the present model to a symmetric two-country two-industry
world, in which each country is an exporter in one industry and an importer in the other. Noticethat settling an AD cases can increase the level of protection only if it raises prices. Then, social
welfare, consisting of consumer surplus and all firms’ profits, must decline. Therefore, even if
each country may have the incentive unilaterally to settle an AD duty for a VER, thesimultaneous pursuit of such policy by both governments harms both countries. Further, since ahigher level of import restriction decreases total industry profit, even if governments were moreconcerned with firm profit than consumer welfare, they would be better off agreeing tomutually ban VERs as a means to settling AD cases. Thus, the GATT/WTO ban on VERs canbe regarded as a commitment device designed to avoid the prisoner’s dilemma outcomes.
There may be even simpler explanation of the GATT ban on VERs in the light of ouranalysis. The importing country is generally worse off under a VER than under AD because
there are no tariff revenues. If AD authorities could act independently to protect firms morethan consumers, then it would be in the interest of the welfare-maximizing government to ban
VERs. In this sense, too, the WTO ban can be regarded as a commitment device.6
Although we base our analysis on Cournot competition for simplicity sake, themechanism driving our results is present in Bertrand competition as well. There, with a price6

We are grateful to a referee for suggesting that we explain the GATT ban on VERs in the light of our model, andproviding this particular explanation.

16
undertaking, investment by one firm still exerts a downward pressure on the export price, whilewith a VER it still relaxes the quota constraint facing the exporter. Therefore, foreign firms
have more of an incentive to engage in FDI under a price undertaking than under a VER. Then
our welfare implications remain valid. Further, since a VER yields a more collusive equilibriumoutcome than a price undertaking as shown by Moore (2005), it makes all the more sense toban VERs.

17
Appendix A
In this appendix we prove Proposition 1 for the case of two foreign firms. Begin with a priceundertaking, p, which we assume is always binding.7 Further, for any impact on imports, p--must exceed the free-trade price, a/4, which we also assume. Foreign Firm 1 maximizes profit!1 = (a – x1 – x2 – xh)x1 ,subject to the constraint p = a – x1 – x2 – xh ≥ p. The best-response function is-

b1 = max {a – x2 – xh – p, (a – x2 – xh)/2}-

Interchanging subscripts yields the best-response function for foreign firm 2. The home firm is
not constrained by the price undertaking. Solving the three-best response functions
simultaneously yields the following symmetric equilibrium export sales per foreign firmx(p) = (a - 2p)/2--

and xh(p) = p for the home firm. Each foreign firm makes the profit

-

(A1)

-

!(p) = p(a - 2p)/2

-

-

-

and the home firm makes(A2)

Π(p) = p2.

--

We now compare an AD duty with a price undertaking that yield the same profit to thedomestic firm. By comparing (3) with (A2), Π(t) = Π(p) implies:-

7

For example, the European Commission closely monitors the price undertakings and imposes penalties to punish
violators (Pauwels and Springael, 2002).

18
(A3)

p = (a + 2t)/4.

-

A comparison with (1) shows that the price is the same with an AD duty. It is easy to check thateach firm’s equilibrium outputs remain identical.
Consider now a VER of size v imposed on the volume of exports from the foreigncountry:x1 + x2 ≤ v.
Assume v < a/2 to have any impact on imports, and that each foreign firm exports up to v/2. It
is easy to check that the Nash equilibrium is(x1, x2, xh) = (v/2, v/2, (a – v)/2).
In equilibrium, industry output is (a + v)/2, and the price is(A4)

p = (a - v)/2;

The profit to each foreign firm is!(v) = v(a - v)/2.
and that to the home firm is(A5)

Π(v) = (a – v)2/4.
We now compare a VER with a price undertaking while keeping the home firm’s profit

constant. By (A2) and (A5), we have
(A6)

(a – v)/2 = p.

-

19
In equilibrium, each foreign firm exports v/2 = (a – 2p)/2, with the home firm producing (a –-v)/2 = p, as under the price undertaking p. This proves the equivalence between a VER and a--

price undertaking.

Appendix B
This appendix contains results we used to draw Figures 2 – 4.
Lemma 1 With AD
A: (E, E) is the unique equilibrium if k > 3t(2a – t)/16.
B: (I, I) is the unique equilibrium if k < 3t(2a – 3t)/16.
C: Either (E, I) or (I, E) is the equilibrium if 3t(2a – 3t)/16 < k <3t(2a – t)/16.
Proof: Result A is true if both these inequalities hold:(B1)(a - 3t)2/16 > a2/16 – k,

(B2)

(a – 2t)2/16 > (a + t)2/16 – k.

The first is written
16k > 3t(2a – 3t),
and the second is written
16k > 3t(2a – t) > 3t(2a – 3t).
Result A is immediate. Result B holds if inequalities (B1) and (B2) hold in reverse. If (B1)holds and (B2) holds in reverse, Result C obtains. Lemma 2 With a price undertaking p ∈(a/4, a/3) corresponding to an AD duty t ∈ (0, a/6)-

20
A. (I, I) is the unique equilibrium if k < t(a – 3t)/4.
B. (E, E) is the unique equilibrium if k > t(a + 2t)/4.
C. Either (E, I) or (I, E) is the equilibrium if t(a – 3t)/4 < k < t(a + 2t)/4.
Proof. Result A is true if both these inequalities hold:(B3)

a2/16 – k > (a – 3p)p,(B4)

p2 – k > p(a – 2p)/2.

--

-

-

-

Using the conversion formula, the first is written
16k < a2 - (a – 6t)(a + 2t) = 4t(a – 3t)
and the second is written
16k < 4t(a + 2t).
Since 4t(a – 3t) < 4t(a + 2t), we obtain Result A. Result B is true if inequalities (B3) and (B4)hold in reverse. If both (B3) fails and (B4) holds, Result C obtains.

Lemma 3: With a price undertaking p ∈(a/3, 5a/12) equivalent to the AD duty t ∈ (a/6, a/3)-

A: (I, I) is the unique equilibrium if k < (7a2 + 36t2)/144.
B: The equilibrium is either (I, I) or (E, E) if k > (7a2 + 36t2)/144.
C: (E, I) and (I, E) are never the equilibrium.
Proof: Result (A) is true if
a2/9 – k > p(a – 2p)/2.

--

Using the conversion formula, this is written

21
k < (7a2 + 36t2)/144.
If the inequality is reversed, (E, E) is also an equilibrium. Results (C) is obvious under theassumption that k < a2/16.

Lemma 4: With a VER v ∈ (a/4, a/2) corresponding to t ∈ (0, a/4), (E, E) is the uniqueequilibrium.
Proof. The lemma is true if
v(a - v)/2 > a2/16 – k.
Since v(a - v)/2 is bounded from below by a2/9 for a/2 > v > a/4, this inequality always holds.
Lemma 5: With a VER v ∈ (a/6, a/4) equivalent to the AD duty t ∈(a/4, a/3)
A: (E, E) is the unique equilibrium if k > (44t2 + 8at +7a2)/72.
B: (I, I) is the unique equilibrium if k < (16t2 - a2)/48.
C: Either (E, E) or (I, I) is the equilibrium if(16t2 - a2)/48 < k < (44t2 + 8at +7a2)/72.
Proof. Result A is true if both these inequalities hold:v(a - v)/2 > (a - v)2/9 - k
a2/16 – k < v(a - v)/3.
Using the conversion formula, the first is writtenk > (44 t2 + 8at - 7a2 )/72

22
and the second is writtenk > (16 t2 - a2 )/48.
Result A follows because (44 t2 + 8at - 7a2 )/72 < (16 t2 - a2 )/48 for a/4 < t < a/3. Result B istrue if both inequalities hold in reverse. Result C is true if both these inequalities hold:v(a - v)/2 > (a - v)2/9 – k
a2/16 – k > v(a - v)/3. Appendix C
We compute home country welfare with a price undertaking that leads to the equilibrium (E, E)
in Case I. Calculations show that the equilibrium consumer surplus is (a – p)2/2, and hence the-home welfare is
WEE(p) = (a – p)2/2 + p2,

-

-

-

where p ≥ a/4. WEE(p) is decreasing for p < a/3 and increasing for p > a/3. Welfare reaches its

-

-

-

-

maximum value of 3a2/8 at p = a/2, at which imports vanish. Given the equivalence, home-country welfare is also decreasing in t < a/6.

23
References
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25

xh
rf
rft
A

N

Nt

rh
C

O

xf

B

Figure 1: The equilibrium outcomes with only one foreign firm

26k(E, I), (I, E)

a2/16
a2/18

(E, E)(I, I)

ta/6

a/3

Figure 2: Equilibrium outcomes under AD

27(I, I), (E, E)

k
a2/16
a2/18
(E, E)(I, I)

a2/48
t(E, I), (I, E)

a/6

a/3

Figure 3: Equilibrium outcomes under price undertakings

28

k
a2/16

(E, E)(E, E), (I, I)(I, I)
a/6

a/4

t
a/3

Figure 4: Equilibrium outcomes under VERs