This document discusses a two-stage game model of tariffs and imperfect international competition. In the first stage, two governments simultaneously choose tariff rates. In the second stage, after observing the tariffs, two firms choose production quantities for domestic and export markets. Using backward induction, the Nash equilibrium of the firm subgame is derived for any given tariffs. The equilibrium tariffs that maximize each government's total welfare are then determined. The subgame-perfect Nash equilibrium has both governments imposing positive tariffs above marginal cost, and firms producing positive quantities for both domestic and export markets.

1. Tariffs and imperfect international
competition
Game Theory
By Eranga Kavirathna
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2.  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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3.  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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4.  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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5.  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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6. 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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7. Tariffs and imperfect international
competition
Firm 1's play for its profit:
Firm 2's play for its profit:
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8. 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
where .
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
where .
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9. 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 .
Firm 1 maximizes (first derivation)
FOC:
Firm 2 maximizes (first derivation)
FOC:
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10. Backward induction: whole game
Both countries know that two firms' best response for any pair
Country 1 maximizes ( )
Plugging what we got into country 1's objective function
FOC:
By symmetry, we also get
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11. 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 .
Given , a Nash equilibrium of the subgame
should satisfy these equations.
Solving these equations gives us
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12. Tariffs and imperfect international
competition
The subgame-perfect Nash equilibrium
The subgame-perfect outcome
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