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1
Tariffs and imperfect international competition
Game Theory
By Eranga Kavirathna
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Two Stage Game Dynamic game with lots of players and imperfect
information.
Imperfect information A player may not know exactly Who has made
What choices when he has an opportunity to make a choice.
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There are more players in this Game. Each country has a
1. government, 2. a firm, and 3. consumers for a firm’s output. (consumers are treated
as passive in this game)
So there are 4 players,1. Two governments 2. Two firms
The players change in the second stage of the game.
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Two governments, 1 and 2, simultaneously choose their tariff rates, denoted by t1, t2.
Firm 1 from country 1 and firm 2 from country 2 produce a homogeneous product for both home consumption and export.
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After observing the tariff rates chosen by the two countries, firm 1 and 2 simultaneously chooses quantities for home consumption and for export, denoted by (h1, e1) and (h2, e2), respectively.
Market price in two countries Pi(Qi)=a–Qi, for i=1, 2.
Q1=h1+e2, Q2=h2+e1.
Both firms have a constant marginal cost c. Each firm pays tariff on export to the other country.
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How high of a tariff should each government impose?It depend on,
Firms behavior (Profit maximization)
Objective of government policy (Total welfare, Maximize TAX revenue, Maximize Home firm revenue or Max. Consumer Surplus)
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Tariffs and imperfect international competition
Firm 1's play for its profit:
12111211212211211 )()]([)]([),,,,,( etehceheahehaehehtt
Firm 2's play for its profit:
21222122122211212 )()]([)]([),,,,,( etehceheahehaehehtt
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Tariffs and imperfect international competition
Country 1's Gov. play for its total welfare: sum of the consumers' surplus enjoyed by the consumers of country 1, firm 1's profit and the tariff revenue
212211211212211211 ),,,,,(
2
1),,,,,( etehehttQehehttW
where 211 ehQ .
Country 2's Gov. play for its total welfare: sum of the consumers' surplus enjoyed by the consumers of country 2, firm 2's profit and the tariff revenue
122211212222211212 ),,,,,(
2
1),,,,,( etehehttQehehttW
where 122 ehQ .
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Backward induction: subgame between the two firms
Here we will find the Nash equilibrium of the subgame between the two firms for any given pair of ) ,( 21 tt .
Firm 1 maximizes (first derivation)
12111211212211211 )()]([)]([),,,,,( etehceheahehaehehtt
FOC: )(
2
1 02
)(2
1 02
221221
2121
tchaetchea
ceahceha
Firm 2 maximizes (first derivation)
21222122122211212 )()]([)]([),,,,,( etehceheahehaehehtt
FOC: )(
2
1 02
)(2
1 02
112112
1212
tchaetchea
ceahceha
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Backward induction: whole game
Both countries know that two firms' best response for any pair ) ,( 21 tt
Country 1 maximizes ( 211 ehQ )
212211211212211211 ),,,,,(
2
1),,,,,( etehehttQehehttW
Plugging what we got into country 1's objective function
)2(3
1)2(
3
1))2(
3
1)(
3
2(
)2(3
1)
3
1)(
3
2()(
3
1)
3
1)(
3
2())(2(
18
1
112221
22112
1
tcattcatttcac
tcatcaatcatcaatca
FOC:
)(3
11 cat
By symmetry, we also get )(3
12 cat
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Backward induction: subgame between the two firms
Here we will find the Nash equilibrium of the subgame between the two firms for any given pair of ) ,( 21 tt .
Given ) ,( 21 tt , a Nash equilibrium ) ) ,( ), ,( ( *2
*2
*1
*1 eheh of the subgame
should satisfy these equations.
)(21
)(21
221
21
tchae
ceah
)(21
)(21
112
12
tchae
ceah
Solving these equations gives us
)2(31
)(31
)2(31
)(31
1*22
*2
2*11
*1
tcaetcah
tcaetcah
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Tariffs and imperfect international competition
The subgame-perfect Nash equilibrium
)2(3
1
)(3
1
,)2(
3
1
)(3
1
),(3
1 ),(
3
1
12
22
21
11*2
*1
tcae
tcah
tcae
tcahcatcat
The subgame-perfect outcome
)(9
1
)(9
4
,)(
9
1
)(9
4
),(3
1 ),(
3
1
*2
*2
*1
*1
*2
*1
cae
cah
cae
cahcatcat
