Optimal tariffs for the co-existence of exportingand non-exporting firms
Kazuhiro Takauchi
Tomomichi Mizuno
October 2022
Discussion Paper No. 2214

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANOptimal tariﬀs for the co-existence of exporting andnon-exporting firms
Kazuhiro Takauchi∗†Tomomichi Mizuno‡

October 16, 2022
Abstract
We consider an optimal tariﬀ policy with the coexistence of less eﬃcient non-exporting andeﬃcient exporting firms. Using a two-way oligopoly trade model with a firm’s quadratic cost,
we show that the optimal tariﬀ rate is U-shaped with respect to the eﬃciency of non-exporting
firms. This implies that under tariﬀ competition, if relative production eﬃciency increases, both
possibilities appear, that is, trade liberalization or protectionism can progress. We also showthat the profit of the non-exporting firm can be greater than that of the exporting firm when
the production eﬃciency of the non-exporting firm is suﬃciently high.

Key words: Optimal tariﬀ; Non-exporting firm; Two-way trade; Quadratic cost
JEL classification: F12; F13∗

Corresponding author: Kazuhiro Takauchi, Faculty of Business and Commerce, Kansai University, 3-3-35
Yamate-cho, Suita, Osaka 564-8680, Japan. E-mail: kazh.takauch@gmail.com; Tel.: +81-6-6368-1817; Fax: +816-6339-7704.
†
Research Fellow, Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe-City,
Hyogo 657-8501, Japan.
‡
Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe-City, Hyogo 657-8501,
Japan. E-mail: mizuno@econ.kobe-u.ac.jp.

1

Introduction

As shown by Brander and Spencer (1984a, b) and Helpman and Krugman (1989), import tariﬀs
can raise national welfare by shifting rents from foreign oligopoly firms to domestic firms. Hence,each country has an incentive to protect its domestic industry, especially the manufacturingindustry. It is well known that in many manufacturing industries, the exporting firm is only apart of the entire industry. In general, within an industry, some firms export, but other firms
only supply goods domestically (Bernard et al., 2007; Freund and Pierola, 2015).1 In addition,it has been empirically demonstrated that exporting firms are more eﬃcient in production thannon-exporting firms (Clerides et al., 1998). If exporting and non-exporting firms coexist, whatis the optimal tariﬀ policy? How does the diﬀerence in production eﬃciency between exportingand non-exporting firms aﬀect the optimal tariﬀ rate?
This study considers a situation in which, using a two-country two-way oligopoly trademodel, non-exporting firms and exporting firms coexist. The non-exporting firm is less eﬃcientthan the exporting one. Furthermore, non-exporting firms supply only domestically, whereas
exporting firms conduct both exports and domestic supply. Under this setting, we show thatthe optimal tariﬀ rate can be higher or lower owing to the degree of ineﬃciency of non-exporting
firms.
A decrease in non-exporting firms’ eﬃciency reduces their output and consumer surplus.
By contrast, the import and domestic supply of exporting firms increases. Hence, whether the
optimal tariﬀ becomes high or low depends on government preference; that is, the government
compensates for a reduction in consumer surplus by using either imports or the domestic supplyof the exporting firm. When the eﬃciency of non-exporting firms is high, it is desirable for welfare
to compensate for a reduction in consumer surplus to use an increase in imports. In this case, thegovernment lowers the tariﬀ rate for a fall of the non-exporting firm’s eﬃciency. Conversely, if the1
For example, according to Bernard et al. (2007), in US data of 2002, the proportion of exporting firms in theentire manufacturing industry was 18%.

1

eﬃciency of a non-exporting firm is low, it is desirable for welfare to compensate for a reduction
in consumer surplus to increase the domestic supply of the exporting firm. The government thenincreases the tariﬀ rate if the non-exporting firm’s eﬃciency decreases. Our analysis contributes
to the literature by showing the specific conditions under which tariﬀ competition results in ahigher tariﬀ, that is, protectionism continues, or in a lower tariﬀ, that is, trade liberalization
advances.
We also show that the profit of a non-exporting firm can be greater than that of an exporting
firm. If the eﬃciency of the non-exporting firm is relatively high, its output is larger than theexports and domestic supply of the exporting firm. Therefore, the profit of the non-exporting
firm is greater than that of the exporting firm. This result indicates that a less eﬃcient domestic
firm is not always inferior to an eﬃcient exporting firm.
This study is related to the following works, which assume a quadratic cost function for thefirms. Choi (2022) considers a competition mode in which exporting firms choose whether priceor quantity is a strategic variable. Khoi et al. (2019) examine the optimal tariﬀ policy undermany-firm intra-industry trade. However, they do not consider the existence of a non-exporting
firm or the diﬀerences in production eﬃciency among firms. In addition, Choi (2022) uses athird-market model, and hence, it does not consider a two-way trade situation.
Although the argument on the optimal tariﬀ under an oligopoly market is in many branches,for example, we can mention the following three strands of literature.2 The first is the argument
on discriminatory tariﬀs (e.g., Choi, 1995; Hwang and Mai, 1991; Liao and Wong, 2006). The
second argument is related to the vertical production structure (e.g., Ara and Ghosh, 2016;
Ishikawa and Lee, 1997; Lahiri and Ono, 1999).3 Third, studies have focused on the relationshipwith public policy, such as privatization (e.g., Chao and Yu, 2006; Long and Stähler, 2009; Pal2

Recently, optimal tariﬀ policy under monopolistic competitive model with firm heterogeneity is also considered. See, for example, Felbermayr et al. (2013).
3
In addition these, Takauchi (2011) considers optimal tariﬀ policy for an importing country within a free tradearea with vertically related markets.

2and White, 1998). Choi (1995) compares the eﬀects of the discriminatory tariﬀ regime and themost favored nation clause on the technology choice of firms. Hwang and Mai (1991) focus on
the relationship between discriminatory tariﬀs and firms’ cost diﬀerences. Liao and Wong (2006)
consider the diﬀerences in the preferable tariﬀ regime between exporting and importing countries.
Ara and Ghosh (2016) examine the optimal tariﬀ under vertical specialization. Ishikawa andLee (1997) find that tariﬀs may harm domestic inputs and final good producers. Lahiri and Ono(1999) consider tariﬀ policies when producers and sellers diﬀer. Chao and Yu (2006) consider
the eﬀect of partial privatization of publicly owned firms or foreign competition on optimal tariﬀ
policy. Long and Stähler (2009) show that optimal tariﬀs can be independent of the degree ofstate ownership. Pal and White (1998) consider the eﬀects of privatization on optimal tariﬀ
and subsidy policies. Although these studies oﬀer interesting results, their purposes and models
diﬀer substantially from ours.
The remainder of this paper is organized as follows. Section 2 presents the model, andSection 3 presents the results. Finally, Section 4 concludes the study.

2

Model

We consider Brander and Krugman’s (1983) two-way trade model. There are two countries, A
and B, and each country i (i = A, B) has two firms that produce homogenous goods. In eachcountry, there are two types of firms: one engaging in only domestic supply, which we call anon-exporting firm. The other, which engages in both domestic supply and export, is called an
export firm. The cost function of non-exporting firm i is γ(qid )2 /2; γ ≥ 1 represents the degreeof production ineﬃciency compared to the exporting firm, and qid is the firm’s output. The coste )2 /2, q e is the domestic supply, and q e represents exportsfunction of exporting firm i is (qiie + qijii
ij(i, j = A, B and i ̸= j). A quadratic cost function is a popular setting and is often employed inoligopoly models (e.g., see Goerke, 2022; Von Weizsacker, 1980).

3

e , where
The inverse demand function in country i (i = A, B) is given by pi = a − qid − qiie − qjie are the imports from country j (j ̸= i).
pi is the price of homogenous good and qji

Each country i imposes a specific tariﬀ, ti for imports. Hence, the profits of the non-exporting
firm, πid , and exporting firm, πie , are given by
πid ≡ pi qid −γ d 2(q ) ,2 ieπie ≡ pi qiie + (pj − tj )qij
−

(1)
e )2
(qiie + qij
.
2

(2)

A simple two-stage game is analyzed. In the first stage, each country independently andsimultaneously decides on the tariﬀ rate ti to maximize total surplus. In the second stage,
non-exporting and exporting firms compete in a Cournot manner in each market. The equilibrium concept is a subgame perfect Nash equilibrium; hence, the game is solved using backward
induction.

3

Results

In the second stage of the game, each firm decides on its quantity. The FOCs for profit maximizae = 0, a−3q e −q e −q d −q e = 0, and a−q e −3q e −q e −q d −t = 0,tion are a−qiie −γqid −2qid −qji
j
ii
ij
i
ji
ii
ij
jj
j

where i, j = A, B and i ̸= j. These FOCs yield the following second-stage output.
qid (ti , tj ) =3a(3γ + 4) + (4γ + 6)ti − (γ + 2)tj,
(3γ + 4)(5γ + 8)

qiie (ti , tj ) =3a(3γ 2 + 7γ + 4) + (9γ 2 + 23γ + 14)ti + (9γ 2 + 28γ + 22)tj,
3(3γ + 4)(5γ + 8)eqij(ti , tj ) =3a(3γ 2 + 7γ + 4) − (3γ + 5)[2(γ + 1)ti + (7γ + 10)tj ].
3(3γ + 4)(5γ + 8)

(3)

In the first stage, the objective function of country i is total surplus.
e
SWi ≡ CSi + πid + πie + ti qji,e )2 /2 is consumer surplus. By using (3), we obtain SW (t , t ). The
where CSi = (qid + qiie + qji
i i j

4

FOCs for the maximization of SWi (ti , tj ) yield the following best-response function BRi (tj ).
ti = BRi (tj ) ≡3a(3γ + 4)(33γ 3 + 133γ 2 + 212γ + 130)1503γ 4 + 9276γ 3 + 21253γ 2 + 21428γ + 8020+

(72γ 4 + 429γ 3 + 913γ 2 + 794γ + 220) tj
for i, j = A, B; i ̸= j.
1503γ 4 + 9276γ 3 + 21253γ 2 + 21428γ + 8020

This BRi (tj ) yields the following result.

Lemma 1. The best response of each country is a strategic complement.

Proof. From BRi (tj ), ∂BRi /∂tj > 0 for i ̸= j. □When the foreign country (country j) raises the tariﬀ rate, the exporting firm of countryi decreases exports and increases domestic supply. At that time, if country i reduces its tariﬀ
rate, imports increase, and the domestic supply of the exporting firm decreases. To improve
national welfare, it is necessary to increase the domestic supply of exporting firms; thus, country
i also raises its tariﬀ rate.
The best-response function of each country, BRi , yields the optimal tariﬀ rate:t∗i =a(33γ 3 + 133γ 2 + 212γ + 130).
159γ 3 + 771γ 2 + 1232γ + 650(4)From (4), we establish Proposition 1.

Proposition 1. I. Suppose 1 ≤ γ ≤ γ ∗ ≃ 1.40847. A decrease in the production eﬃciency of the
non-exporting firm (i.e., an increase in γ), lowers tariﬀ rate. II. Suppose γ > γ ∗ . A decrease in
the production eﬃciency of the non-exporting firm increases the tariﬀ rate.

Proof. Diﬀerentiating (4) with respect to γ, we obtain∂t∗i8a(537γ 4 + 1737γ 3 + 343γ 2 − 3445γ − 2795)=.
∂γ(159γ 3 + 771γ 2 + 1232γ + 650)2
By solving ∂t∗i /∂γ ≥ 0 for γ, we find that ∂t∗i /∂γ ≥(<)0 if γ ≥(<)γ ∗ ≃ 1.40847. □5

Proposition 1 states that a tariﬀ race can result in both lower and higher tariﬀ rates. An
increase in γ reduces the output of non-exporting firms and consumer surplus. This reduction
in consumer surplus is partly compensated by an increase in the domestic supply of exporting
firms and foreign firms’ exports. Whether the government increases or decreases its tariﬀ ratedepends on whether the government prefers to rely on either an increase in the domestic supplyof the exporting firm or an increase in exports of the foreign firm.
When γ is small, the output of the non-exporting firm significantly decreases as γ increases.
Then, if the government compensates for the loss of consumer surplus using only an increasein the domestic supply of the exporting firm, because the marginal cost of the exporting firm
increases widely, it is not desirable for welfare. In this case, lowering the tariﬀ rate to promote
exports of the foreign firm, and not increasing the exporting firm’s marginal cost, is optimal for
welfare.
By contrast, when γ is large, the consumer surplus does not so decrease, even if γ increases.
Hence, even though the government compensates for the loss of consumer surplus mainly by increasing the domestic supply of the exporting firm, the firm’s marginal cost does not so increase.
Therefore, the government increases the tariﬀ rate.
The equilibrium output is given by:qid∗ =

2a(3γ + 5)(17γ + 26),
159γ 3 + 771γ 2 + 1232γ + 650(5)qiie∗ =a(45γ 3 + 187γ 2 + 266γ + 130)6aγ(γ + 3)(2γ + 3)
e∗
, qij
=
.
3
2
3
159γ + 771γ + 1232γ + 650
159γ + 771γ 2 + 1232γ + 650(6)(5) and (6) yield the following result:e∗ ) − q d∗ > 0 for all γ ≥ 1.
Lemma 2. I. (qiie∗ + qijie∗ . (ii) If γ < γ < γ ≃ 7.43869, q e∗ > q d∗ >II. (i) If γ < γ1 ≃ 1.20542, then qid∗ > qiie∗ > qij
12iiie∗ . (iii) If γ > γ , then q e∗ > q e∗ > q d∗ .
qij
2
ii
ij
i

Proof. See Appendix.
6

The exporting firm’s productivity is higher than that of the non-exporting firm, so the total
sales (total amount of domestic supply and exports) of the exporting firm is larger than thatof the non-exporting firm. However, the individual outputs are diﬀerent. The exporting firmmust pay a tariﬀ to export; hence, the domestic supply is always larger than its exports. In
addition, the cost of the tariﬀ exists, so the domestic supply of the exporting firm can be smaller
than that of the non-exporting firm. If γ is suﬃciently small, the non-exporting firm is highly
eﬃcient; hence, its output is larger. Thus, when γ is suﬃciently small (γ < γ1 ), the output of
the non-exporting firm is the largest among all outputs. On one hand, because the output of
the non-exporting firm decreases as γ increases, the output of the non-exporting firm is smaller
than that of the exporting firm when γ is suﬃciently large (γ > γ2 ).
From (1) and (2), the equilibrium profits are:πid∗ =

2a2 (γ + 2)(3γ + 5)2 (17γ + 26)2,
(159γ 3 + 771γ 2 + 1232γ + 650)2(7)πie∗ =

3a2 (2529γ 6 +21242γ 5 +73601γ 4 +134364γ 3 +136548γ 2 +73840γ+16900).
2(159γ 3 + 771γ 2 + 1232γ + 650)2(8)

From (7) and (8), Proposition 2 is established.

Proposition 2. I. If a non-exporting firm is relatively eﬃcient, that is, γ < γ3 ≃ 1.04676, thenthe profit of the non-exporting firm is larger than that of the exporting firm. Otherwise, the profitof the non-exporting firm is lower than that of the exporting firm.
II. A decrease in the production eﬃciency of the non-exporting firm decreases its profit, but
increases the profit of the exporting firm.

Proof. See Appendix.
From Lemma 2, when γ is suﬃciently small, the output of the non-exporting firm is thelargest among all other outputs. The non-exporting firm’s profit increases as γ decreases. On
the one hand, the exporting firm must pay a tariﬀ, so the export is less eﬃcient than domestic

7

supply. Production ineﬃciency is a negative factor for exporting firms. Hence, when γ is
suﬃciently small, the profit of the non-exporting firm is greater than that of the exporting firm.
The output of non-exporting firm i decreases as γ increases. Then, because the strategic
substitute works, exporting firm i’s domestic supply and foreign exporting firm j’s exports
increase. The situation is the same in country j, so the exports of exporting firm i increase.
Hence, if γ increases, the profit of the non-exporting firm decreases, whereas the profit of theexporting firm increases.
The equilibrium consumer surplus and total surplus are
CSi∗ =a2 (γ + 3)2 (57γ 2 + 172γ + 130)2,2(159γ 3 + 771γ 2 + 1232γ + 650)2(9)SWi∗ =

2a2 (3γ + 5)(17γ + 26)(57γ 4 + 406γ 3 + 1118γ 2 + 1389γ + 650).
(159γ 3 + 771γ 2 + 1232γ + 650)2(10)

From (9) and (10), the following result is obtained:

Proposition 3. A decrease in the production eﬃciency of the non-exporting firm reduces consumer surplus and total surplus.

Proof. See Appendix.
Proposition 3 is thus intuitive. An increase in γ worsens productivity; thus, aggregate
output decreases. Hence, consumer surplus decreases as γ increases. Furthermore, because thereduction eﬀect of consumer surplus is dominant, the total surplus decreases as γ increases.

4

Conclusion

Using a two-country oligopoly model, we consider the eﬀects of technology diﬀerences between
non-exporting and exporting firms on optimal tariﬀ policy. The non-exporting firm is lesseﬃcient than the exporting firm, and conducts only domestic production. By contrast, exporting
firms engage in both domestic production and exports. In this situation, we show that, according8

to the degree of eﬃciency of non-exporting firms, the optimal tariﬀ rate becomes low or high.
When the production eﬃciency of a non-exporting firm is high (low), the optimal tariﬀ rate
decreases (increases), as its eﬃciency decreases. We believe that our analysis contributes to theliterature on the optimum tariﬀ argument by showing that, under tariﬀ competition, a low (orhigh) tariﬀ is achieved and trade liberalization (or protectionism) continues.

Acknowledgements
We thank Noriaki Matsushima for his helpful comments. Takauchi acknowledges the financial
support from the Japan Society for the Promotion of Science (JSPS), KAKENHI Grant Number
20K01646. All errors are our own.

Appendix. Proofs
Proof of Lemma 2e∗ , and q d∗ , we have
I. From qiie∗ , qijie∗(qiie∗ + qij) − qid∗ =a(57γ 3 + 139γ 2 − 6γ − 130)> 0.
159γ 3 + 771γ 2 + 1232γ + 650

II. Some algebra yield)
33γ 3 + 133γ 2 + 212γ + 130> 0.
159γ 3 + 771γ 2 + 1232γ + 650()
5a 9γ 3 + 17γ 2 − 12γ − 26e∗d∗qii − qi =,159γ 3 + 771γ 2 + 1232γ + 650ae∗qiie∗ − qij=(

where qid∗ − qiie∗ ≥(<)0 for γ ≤(>)γ1 ≃ 1.20542.
e∗qid∗ − qij=−()
4a 3γ 3 − 12γ 2 − 68γ − 65,159γ 3 + 771γ 2 + 1232γ + 650e∗ ≥(<)0 for γ ≤(>)γ ≃ 7.43869. These imply Lemma 2. □where qid∗ − qij2

Proof of Proposition 2

9

I. A simple algebra yields
πid∗ −πie∗ = −a2 (7587γ 6 +53322γ 5 +133491γ 4 +110768γ 3 −78508γ 2 −185120γ−84500).
2(159γ 3 + 771γ 2 + 1232γ + 650)2

By numerically solving πid∗ − πie∗ ≥ 0 for γ, we have γ ≤ γ3 ≃ 1.04676.
II. From the diﬀerentiation of profits:∂πid∗
2a2 (3γ+5)(17γ+26)(8109γ 5 +70866γ 4 +247863γ 3 +433528γ 2 +378950γ+132340)=−< 0,(159γ 3 +771γ 2 +1232γ+650)3∂γ∂πie∗
24a2 (32640γ 7 +339711γ 6 +1517619γ 5 +3746887γ 4 +5477452γ 3 +4693455γ 2 +2151370γ+397150)=> 0.
(159γ 3 +771γ 2 +1232γ+650)3∂γ
These imply Proposition 2. □Proof of Proposition 3.
Diﬀerentiating CSi∗ and SWi∗ with respect to γ, we obtain:
10a2 (γ+3)(57γ 2 +172γ+130)(1059γ 4 +6498γ 3 +15037γ 2 +15548γ+6058)
∂CSi∗
=−
< 0,
∂γ
(159γ 3 + 771γ 2 + 1232γ + 650)3
7 +4079325γ 6 +23685669γ 5 +74023721γ 4 +135748078γ 3286929γ
2a2 
2 + 87224540γ + 21970000∗+146920410γ
∂SWi
=−
< 0.
∂γ
(159γ 3 + 771γ 2 + 1232γ + 650)3Hence, Proposition 3 holds. □References[1] Ara, T. and A. Ghosh (2016) Tariﬀs, vertical specialization, and oligopoly. European Economic Review, 82, 1-23.
[2] Bernard, A., B. Jensen, S. J. Redding, and P. K. Schott (2007) Firms in international trade.
Journal of Economic Perspectives, 21(3), 105-130.
[3] Brander, J. A. and P. R. Krugman (1983) A ‘reciprocal dumping’ model of international
trade. Journal of International Economics, 15, 313-321.

10

[4] Brander, J. A. and B. Spencer (1984a) Trade warfare: Tariﬀs and cartels. Journal of International Economics, 16, 227-242.
[5] Brander, J. A. and B. Spencer (1984b) Tariﬀ protection and imperfect competition. In H.
Kierzkowski (eds.) Monopolistic Competition and International Trade (Oxford: Clarendon
Press).
[6] Chao, C. C., and Yu, E. S. (2006) Partial privatization, foreign competition, and optimum
tariﬀ. Review of International Economics, 14(1), 87-92.
[7] Choi, J. P. (1995) Optimal tariﬀs and the choice of technology: Discriminatory tariﬀs vs.
the ‘Most Favored Nation’ clause. Journal of International Economics, 38(1-2), 143-160.
[8] Choi, K. (2022) Uniform versus discriminatory tariﬀs in increasing marginal costs and endogenous competition. SSRN Paper.
[9] Clerides, S. K., S. Lach, and J. R. Tybout (1998) Is learning by exporting important? Microdynamic evidence from Colombia, Mexico, and Morocco. Quarterly Journal of Economics,
113(3), 903-947.
[10] Freund, C. and M. D. Pierola (2015) Export superstars. Review of Economics and Statistics,
97(5), 1023-1032.
[11] Goerke, L. (2022) Partisan competition authorities, Cournot-oligopoly, and endogenous
market structure. Southern Economic Journal (forthcoming).
[12] Helpman, E. and Krugman P. R. (1989) Trade Policy and Market Structure, The MIT
Press.
[13] Hwang, H. and C.-C. Mai (1991) Optimum discriminatory tariﬀs under oligopolistic competition. Canadian Journal of Economics, 24(3), 693-702.

11

[14] Ishikawa, J., and Lee, K. D. (1997) Backfiring tariﬀs in vertically related markets. Journalof International Economics, 42(3-4), 395-423.
[15] Khoi, L. X., Trung, N. D., and Van, L. X. (2019) Oligopoly model and its applicationsin international trade. In International Econometric Conference of Vietnam (pp. 296-310).
Springer, Cham.
[16] Lahiri, S. and Y. Ono (1999) Optimal tariﬀs in the presence of middlemen. Canadian
Journal of Economics, 32(1), 55-70.
[17] Long, N. V. and F. Stähler (2009) Trade policy and mixed enterprises. Canadian Journalof Economics, 42(2), 590-614.
[18] Pal, D., and White, M. D. (1998) Mixed oligopoly, privatization, and strategic trade policy.
Southern Economic Journal, 65(2), 264-281.
[19] Takauchi, K. (2011) Rules of origin and international R&D rivalry. Economics Bulletin,
31(3), 2319-2332.
[20] Von Weizsacker, C. C. (1980) A welfare analysis of barriers to entry. Bell Journal of Economics, 11(2), 399-420.

12

Online Appendix (Not for Publication) on“Optimal tariﬀs for the coexistence of exporting andnon-exporting firms”
Here, we present supplementary information related to the analysis in this study.
Lamma A1. A suﬃcient condition for equilibrium to be asymptocically stable is satisfied, that
is, |∂BRA /∂tB ||∂BRB /∂tA | < 1 for all γ ≥ 1.
Proof. The best response function of each country, BRi (i = A, B), yields
∂BRA
∂tB
=−

∂BRB
−1
∂tA

3(3γ + 4)(5γ + 8)(159γ 3 + 771γ 2 + 1232γ + 650)(315γ 3 + 1437γ 2 + 2134γ + 1030)(1503γ 4 + 9276γ 3 + 21253γ 2 + 21428γ + 8020)2< 0.

Hence, Lemma A1 holds. □SOC for the welfare maximization
From the second stage outcome of the game, we obtain∂ 2 SWi
1503γ 4 + 9276γ 3 + 21253γ 2 + 21428γ + 8020
< 0,=−9(3γ + 4)2 (5γ + 8)2
∂t2i
and
( 2
)( 2
)
)( 2
) ( 2
∂ SWA
∂ SWB
∂ SWB
∂ SWB
−
∂tB ∂tA∂tA ∂tB
∂t2A
∂t2B
(
)(
)
159γ 3 + 771γ 2 + 1232γ + 650 315γ 3 + 1437γ 2 + 2134γ + 1030=
> 0.
27(3γ + 4)3 (5γ + 8)3
Thus, the SOC is satisfied.

13