Optimum Tariff and Finite Sequential Trade...

1

Optimum Tariff and Finite Sequential Trade War Final Project of ECON 567

Xin Chen

Spring, 2014

Introduction This purpose of this project is to demonstrate the effect of unilateral optimum tariff and a

simulation of finite sequential tariff war under a two country two product HOS (Heckscher-Ohlin-

Samuelson) model using GAMS. The simulation of the tariff war evaluated symmetric and asymmetric

setting of trading countries. It also studied the existence of first mover advantage in such tariff war. This

model adapted from "two country version of the HOS model with trade taxes", as described in Chapter 19

of Gilbert and Tower: Introduction to Numerical Simulation for Trade Theory and Policy.

Outlines

Part I 1. Solve the initial model of two countries with no tariff

2. Solve the optimal US import tariff to maximize US utility, given no tariff of UK

3. Lerner Symmetry Theorem - Solve the optimal US export tax/subsidy to maximize US utility,

given no tariff of UK

4. Lerner Neutrality Theorem - US sets both import and export taxes at 10%, given no tariff of UK

5. Factor price equalization theorem

6. Stolper-Samuelson Theorem - Plot US Import Tariff on Beer(T) vs. Utility(U), Consumptions(C),

Export(X) and Production(Q), Given Zero Tax in UK

7. Sensitivity Analysis on

a. Beta ('Auto','US')

b. FBAR (‘K’,’US’) (US Capital Endowment).

Part II Simulate a retaliatory trade war between US and UK, with each country imposing optimal trade taxes in

turn.

8. Finite Sequential Trade War (US moves in odd periods, UK moves in even periods)

a. US starts first, set the optimal tariff, given UK tax fixed.

b. UK then sets its optimal tariff, given US tariff rate from previous step.

c. Repeat for 10 rounds

9. Finite Sequential Trade War (US moves in odd periods, UK moves in even periods, adjustment of

share parameters in utility of US)

2

10. Finite Sequential Trade War (US moves in odd periods, UK moves in even periods, adjustment of


11. Finite Sequential Trade War (UK moves in odd periods, US moves in even periods, adjustment of


Model The model employed in this paper is the two country version of the HOS (Heckscher-Ohlin-

Samuelson) model with trade taxes.

In this model, we have two countries, US and UK. Each produces two goods, auto and beer, using

identical CES production function. Auto is capital intensive and beer is labor intensive. US has relatively

higher ratio of capital to labor.

Assumptions

1. Endowment factors

2. Flexible factor prices

3. Flexible exchange rate

4. Production function with constant elasticity of technological substitution(CES)

Sets J = {K, L}

I = {Auto, Beer}

D = {US, UK}

3

Table 1. Parameters

Description Notation Initial Value

Shift parameter in utility 𝜶𝒅 US = 2

UK = 2

Share parameter in utility

𝜷𝒊𝒅 Auto.US = 0.5

Auto.UK = 0.5

Beer.US = 0.5

Beer.UK = 0.5

Initial domestic prices 𝑷𝑶𝒊𝒅 1

Initial world prices 𝑷𝑾𝑶𝒊 1

Initial trade

𝑿𝑶𝒊𝒅∗ (𝑋𝑂𝑖𝑑 = 𝑄𝑂𝑖𝑑 − 𝐶𝑂𝑖𝑑)

Auto.US = 62.5

Auto.UK = -62.5

Beer.US = -62.5

Beer.UK = 62.5

Initial utility levels 𝑼𝑶𝒊𝒅 US = 275

UK = 275

Initial consumption levels 𝑪𝒊𝒅 137.5

Shift parameters in production

𝜸𝒊𝒅 Auto.US = 1.676

Auto.UK = 1.676

Beer.US = 1.676

Beer.UK = 1.676

Share parameters in production

𝜹𝒋𝒊𝒅 K.Auto.US = 0.777

K.Auto.UK = 0.777

K.Beer.US = 0.223

K.Beer.UK = 0.223

L.Auto.US = 0.223

L.Auto.UK = 0.223

L.Beer.US = 0.777

L.Beer.UK = 0.777

Elasticity parameters in production 𝝆𝒊𝒅 0.1

Factor Endowments

�̅�𝒋𝒅 K.US = 175

K.UK = 100

L.US = 100

L.UK = 175

Initial output levels

𝑸𝑶𝒊𝒅 Auto.US = 200

Auto.UK = 75

Beer.US = 75

Beer.UK = 200

Initial factor prices 𝑹𝑶𝒊𝒅 1

Initial factor use levels

𝑭𝑶𝒋𝒊𝒅 K.Auto.US = 160

K.Auto.UK = 60

K.Beer.US = 15

K.Beer.UK = 40

L.Auto.US = 40

L.Auto.UK = 15

L.Beer.US = 60

L.Beer.UK = 160

Initial GDP 𝑮𝑫𝑷𝑶𝒅 US = 275

UK = 275

4

Table 2. Variables

Description Notation Numbers

Utility Indices 𝑼𝒊 2

Tax on import/export 𝑻𝒊𝒅 3+1

Prices 𝑷𝒊𝒅 4

Consumption Levels 𝑪𝒊𝒅 4

Output Levels 𝑸𝒊𝒅 4

Factor Prices 𝑹𝒋𝒅 4

Factor Use Level 𝑭𝒋𝒊𝒅 8

World Prices 𝑷𝑾𝒊 2

Trade Levels 𝑿𝒊𝒅 4

Gross Domestic Product 𝑮𝑫𝑷𝒅 2

Note: Variables in red are endogenous.

Table 3. Equations

Description Equation #

Utility functions 𝑈𝑑 = 𝛼 ∏ 𝐶𝑖

𝛽𝑖𝑑

∀𝑖∈𝑰

∀𝑑 ∈ 𝑫 2

Demand functions 𝐶𝑖𝑑 =

𝛽𝑖𝑑 𝐺𝐷𝑃𝑑

𝑝𝑖,𝑑 ∀ 𝑖 ∈ 𝑰, ∀ 𝑑 ∈ 𝑫 4

Open economy

material balance

𝑋𝑖𝑑 = 𝑄𝑖𝑑 − 𝐶𝑖𝑑 ∀ 𝑖 ∈ 𝑰, ∀ 𝑑 ∈ 𝑫 4

International market

clearing

∑ 𝑋𝑖𝑑 = 0

∀𝑑∈𝑫

∀ 𝑖 ∈ 𝑰 2

International price

arbitrage

𝑃𝑖𝑑 = (1 + 𝑇𝑖𝑑) 𝑃𝑊𝑖 ∀ 𝑖 ∈ 𝑰, ∀ 𝑑 ∈ 𝑫 4

Production functions

𝑄𝑖𝑑 = 𝛾𝑖𝑑 [ ∑ 𝛿𝑗𝑖𝑑𝐹𝑗𝑖𝑑𝜌𝑖𝑑

∀𝑗∈𝑱

]

1𝜌𝑖𝑑

∀ 𝑖 ∈ 𝑰, ∀ 𝑑 ∈ 𝑫

(e.g. 𝑄 = 𝛾[𝛿𝐾𝜌 + (1 − 𝛿)𝐿𝜌]1

𝜌)

4

Resource constraints

𝐹𝑗𝑑̅̅ ̅̅ = ∑ 𝐹𝑗𝑖𝑑

∀𝑖∈𝑰

∀ 𝑗 ∈ 𝑱, ∀ 𝑑 ∈ 𝑫

(e.g. 𝐾𝑈𝑆 = 𝐾𝐴𝑢𝑡𝑜,𝑈𝑆 + 𝐾𝐵𝑒𝑒𝑟,𝑈𝑆)

4

5

Factor demand

functions

𝑅𝑗𝑖𝑑 = 𝑃𝑖𝑑𝑄𝑖𝑑 [ ∑ 𝛿𝑗𝑖𝑑𝐹𝑗𝑖𝑑𝜌𝑖𝑑

∀𝑗∈𝑱

]

−1

𝛿𝑗𝑖𝑑𝐹𝑗𝑖𝑑𝜌𝑖𝑑−1

∀ 𝑗 ∈ 𝑱, ∀ 𝑖 ∈ 𝑰, ∀ 𝑑 ∈ 𝑫

8

Gross domestic

products

𝐺𝐷𝑃𝑑 = ∑ 𝑃𝑖𝑑𝑄𝑖𝑑

∀𝑖∈𝑰

− ∑ 𝑃𝑊𝑖 𝑇𝑖𝑑 𝑋𝑖𝑑

∀𝑖∈𝑰

∀ 𝑑 ∈ 𝑫 2

Given our equations of open economy material balance, X(I) is defined such that exports are

positive and imports are negative. Consumer and firm behavior in response to domestic prices is reflected

by demand and consumption functions.

In the original model, both countries share the identical production function and utility function.

The utility function is in Cobb-Douglas form with sum of exponents equaling to one. The share

parameters in utility functions are all equal to 0.5, which means that autos and beers each take half of the

total expenditure for both countries. The production function has constant elasticity of substitution. The

CES function has a elasticity parameter 𝜌 equals 0.1, while the elasticity of substitution equals 1/(1 − 𝜌).

The function converges to the Cobb-Douglas case as 𝜌 → 0, is linear when 𝜌 = 1 and approaches the

Leontief (min) function as 𝜌 → −∞. So the initial value of 𝜌 = 0.1 is more close to a Cobb-Douglas

form.

The only difference between the two countries is their factor endowments. The ratio of capital to

labor for US is reciprocal of that for UK. The sums of K and L for both countries are the same. In other

words, this is a symmetric model of two countries with same size. Later on in this paper, we will modify

some parameters and the size of the countries to conduct sensitivity analysis and simulation of sequential

tariff war.

Results

Part I

The results of scenario 1-5 are shown in table 4 and 5. Table 4 shows the results of optimal tariffs

and demonstration of Lerner Symmetry Theorem. Table 5 illustrates factor price equalization theorem.

Each scenario is shown in a column highlighted with different color. Rows correspond to specific

variables. Variables, such as T, have more than one dimension (country and good). So the values are

6

shown in a matrix with country labeled horizontally and other dimension vertically. Cells with missing

value in T section represent zero tariff.

In scenario 1, we can see a symmetry results with identical utility and consumption level under

free trade. The utility is 275 for both countries. US exports autos and UK exports beers. In our model, US

endows relatively more capital and UK endows more labor. Auto is capital intensive and beer is labor

intensive. This is consistent with Heckscher-Ohlin Theorem which states that if tastes are identical and

homothetic in the two countries, each country will export the commodity which uses intensively the factor

with which it is relatively well endowed.

Scenario 2 and 3 evaluates the optimal tariff and export tax for US to maximize its utility, given

free trade in UK. We can see that both import tariff and export tax can achieve same level of utility for

US. US has higher utility level (from 275 to 278.24) after it imposing optimal tariff/subsidy at the

expense of UK’s welfare (from 275 to 269.58). This demonstrates the Lerner Symmetry Theorem which

says that an across-the-board import tariff is identical to an across-the-board export tariff assuming that

the exchange rate always adjusts to maintain balance of payments equilibrium, and no individual holds

any assets in the other country. The joint utility of both countries, however, drops after the tariff (from

550 to 547.82). This shows that a unilateral optimal tariff is not the best way to transfer wealth.

Transferring purchasing power by cash while maintain free trade is a better way in such transfer problem.

Scenario 4 simply shows that Lerner Neutrality Theorem holds in such two country general

equilibrium model. In this scenario, US imposes 10% import tariff and 10% export tax. Comparing

scenario 1, we can see that the only differences are the values in red. Notice domestic good prices and

factor prices in US increase 10% while all real variables hold constant. This is consistent with Lerner

Neutrality Theorem which states that a tariff on all imports combined with an identical subsidy on all

exports will have no economic impact. The increasing in prices are due to the exchange rate adjustment.

Relative prices, however, do not change. GDPs are calculated based on domestic prices, so US’s GDP

increases 10% while UK’s remains constant.

7

Table 4. Results of Evaluation of Optimal Tariff

Variables

1. No Tariff 2. US Opt.

Tariff

3. US Opt.

Export Tax

4.Lerner

Neutrality

US UK US UK US UK US UK

U 275 275 278.24 269.58 278.24 269.58 275 275


T Auto -0.19 0.1

Beer 0.23 0.1


P Auto 1 1 1.00 1.00 0.81 1.00 1.1 1

Beer 1 1 1.11 0.90 0.90 0.90 1.1 1


C Auto 137.5 137.5 146.49 127.94 146.49 127.94 137.5 137.5

Beer 137.5 137.5 132.12 142.01 132.12 142.01 137.5 137.5


Q Auto 200 75 185.79 88.64 185.79 88.64 200 75

Beer 75 200 88.50 185.63 88.50 185.63 75 200


R K 1 1 0.97 1.04 0.79 1.04 1.1 1

L 1 1 1.15 0.87 0.93 0.87 1.1 1


F

K Auto 160 60 154.38 68.20 154.38 68.20 160 60

K Beer 15 40 20.62 31.80 20.62 31.80 15 40

L Auto 40 15 31.88 20.69 31.88 20.69 40 15

L Beer 60 160 68.12 154.31 68.12 154.31 60 160

Auto Beer Auto Beer Auto Beer Auto Beer

PW 1 1 1.00 0.90 1.00 0.90 1 1


X Auto 62.5 -62.5 39.29 -39.29 39.29 -39.29 62.5 -62.5

Beer -62.5 62.5 -43.62 43.62 -43.62 43.62 -62.5 62.5


GDP 275 275 292.99 255.87 238.04 255.87 302.5 275

Table 5 shows that the factor price equalization theorem holds in such model setting. Scenario 1

is listed here for comparison. The theorem relies on the assumption that there exists free international

trade, 2 factors of production, 2 goods, identical production functions in both countries, only one

diversification cone at equilibrium prices, and neither country is specialized. If these assumptions hold,

then factor prices will be equalized absolutely worldwide.

Given the assumptions hold, the first case in scenario 5 has the results which are consistent with

factor price equalization theorem. In this case, the US endowment of capital changes from 175 to 100.

8

Other settings are the same as initial. The results indicate that the capital prices rise in both countries from

1 to 1.06, and labor prices drops from 1 to 0.79. The relative prices change is due to the decrease of

aggregate capital endowment so that the relative scarcity of capital raise its price. Also notice that US

now becomes smaller in terms of the endowment size, so its optimal utility under free trade becomes

smaller. UK, on the other hand, also suffers because the relative price of beer drops as a consequence of

the capital shrink. Notice that although the relative factor price changes, the factor prices are equal

between two countries. So the factor equalization theorem is preserved in this case.

In the second case of scenario 5, the factor of elasticity of substitution in production become

0.00001 in both countries, which means the production function is almost Cobb-Douglas. The US capital

endowment becomes 9999 and US labor endowment becomes 10. Under such setting, the factor price

equalization theorem no longer holds. The factor prices between two countries are no longer equal. A

speculation is that this setting has more than one diversification cone at equilibrium prices, so that the

assumptions of the theorem are no longer supported.

9

Table 5. Demonstration of factor price equalization theorem

Variables 1. No Tariff 5. Factor Price Equalization Theorem

US UK US UK US UK

U 275 275 202.37 267.06 701.26 568.02

US UK US UK US UK

T Auto

Beer

US UK US UK US UK

P Auto 1 1 1.00 1.00 1.00 1.00

Beer 1 1 0.84 0.84 1.34 1.34

US UK US UK US UK

C Auto 137.5 137.5 92.55 122.14 237.52 171.46

Beer 137.5 137.5 110.63 145.99 191.24 125.18

US UK US UK US UK

Q Auto 200 75 116.95 97.74 375.33 33.66

Beer 75 200 81.46 175.15 74.47 241.95

US UK US UK US UK

R K 1 1 1.06 1.06 0.45 0.89

L 1 1 0.79 0.79 4.05 1.45

US UK US UK US UK

F

K Auto 160 60 87.50 73.13 387.58 33.66

K Beer 15 40 12.50 26.87 28.13 66.34

L Auto 40 15 30.44 25.44 11.33

L Beer 60 160 69.56 149.56 8.42 175.61

Auto Beer Auto Beer Auto Beer

PW 1 1 1.00 0.84 1.00 1.34

US UK US UK US UK

X Auto 62.5 -62.5 24.40 -24.40 137.81 -137.81

Beer -62.5 62.5 -29.16 29.16 -116.77 116.77

US UK US UK US UK

GDP 275 275 185.11 244.28 475.05 342.93

Initial Factor

Endowment:

K.US = 175

K.UK = 100

L.US = 100

L.UK = 175

Factor Endowment:

K.US = 100

K.UK = 100

L.US = 100

L.UK = 175

RHO(I,D)=0.00001

Factor

Endowment:

K.US = 9999

K.UK = 100

L.US = 10

L.UK = 200

In the following five graphs, we plot US utility, consumptions, productions and trade volumes to

see the effect of US tariff rate on these variables, given free trade in UK. We can see that the utility curve

is concave and the optimal tariff rate which maximize utility is around 0.23, which is consistent with the

10

value in scenario 1 of table 4. As US tariff of beer become higher, the US relative prices of beer to autos

rises. As a result, the US beer consumptions drops and auto consumptions rises. We also notice that the

US price of K drops and price of L rises. This result is consistent with Stopler-Samuelson Theorem which

predicts that assuming a country is not specialized, an increase in the relative price of a good

unambiguously raises the real reward of the factor used intensively in producing that commodity and

lowers the real reward of the other factor. In our case, rising beer price rewards the labor as price of L

rises and punish the capitalist.

260

265

270

275

280

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1

US IMPORT TARIFF ON BEER

US UTILITY

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1


US GOOD AND FACTOR PRICES

US Auto Price US Beer Price US Price of K US Price of L

11

Sensitivity Analysis

We conducted two sensitivity analysis on auto share parameter of US utility Beta('Auto','US') and

US capital endowment FBAR(‘K’, ‘US’). In both studies, US sets optimal tariff to maximize its utility

while UK is under free trade.

50

70

90

110

130

150

170

190

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1

US PRODUCTIONS AND CONSUMPTIONS

US Auto Consumption US Beer Consumption

US Auto Production US Beer Production

-80

-60

-40

-20

0

20

40

60

80

0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2


TRADE VOLUMN

US Beer Import US Auto Import

12

The following four graphs are results of sensitivity analysis on auto share parameter of US utility,

Beta('Auto','US'). The utility function is in Cobb-Douglas form, so higher Beta('Auto','US') represents

higher proportion of the expenditure on auto consumption. We plot the endogenous variables with

Beta('Auto','US') from 0.3 to 1. Keep in mind that the initial value of Beta('Auto','US') in our original

model is 0.5.

Notice that US utility is a convex function of Beta('Auto','US'). The minimum is at around

Beta('Auto','US') = 0.45. As US has larger share of expenditure on autos, it consumes more autos and less

beers, so both its import and export decrease. Beer relative price decreases as a respond to the lower

demand, and US produces more auto and less beer accordingly. UK’s utility slightly goes down with the

increasing of Beta('Auto','US').

250

300

350

400

450

500

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

BETA('BEER','US')

UTILITY

US Utility UK Utility

-0.2

0.3

0.8

1.3

0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1BETA('BEER','US')

TARIFF AND PRICES

T* US Auto Price US Beer Price

13

The second sensitivity analysis is on US capital endowment FBAR(‘K’, ‘US’). We plot the effect

of US capital endowment growth on endogenous variables. The results show that both US and UK benefit

from such capital expansion, but US is significantly better off. US beer production remains relatively

constant while its production of auto booms. The US relative price of beer grows significantly, which is

due to the expansion on auto production. It is not a surprise that US price of K drops while US price of

labor increases due to the relative scarcity of labor. An interesting result is that the US optimal tariff

0

50

100

150

200

250

300

0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1

BETA('BEER','US')

CONSUMPTION AND PRODUCTION



-80

-60

-40

-20

0

20

40

60

80

0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1

BETA('BEER','US')

TRADE VOLUMN


14

revenue does not rise as fast as its utility increases, because most of its gain on utility comes from the

increasing consumption on autos rather than tariff revenue and growth of consumption on beer.

250

270

290

310

330

350

370

390

1 7 0 1 9 0 2 1 0 2 3 0 2 5 0 2 7 0 2 9 0 3 1 0

US K ENDOWMENT

UTILITY


0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1 7 0 1 9 0 2 1 0 2 3 0 2 5 0 2 7 0 2 9 0 3 1 0

US K ENDOWMENT

PRICES OF GOODS AND FACTORS

US Auto Price US Beer Price US Price of K US Price of L

15

0

50

100

150

200

250

300

350

1 7 0 1 9 0 2 1 0 2 3 0 2 5 0 2 7 0 2 9 0 3 1 0

US K ENDOWMENT

US PRODUCTION AND CONSUMPTION



-80

-60

-40

-20

0

20

40

60

80

1 7 0 1 9 0 2 1 0 2 3 0 2 5 0 2 7 0 2 9 0 3 1 0

US K ENDOWMENT

US IMPORTS


16

Part II Finite Sequential Tariff War

Traditional textbook theory follows the work of Johnson (1953), considering the case of two

countries that attempt to maximize some form of social welfare by imposing a tariff to shif t the offer

curves. A Cournot behavioral assumption is typically used to explain the reaction to a change in a tariff.

The well-known result is a tariff war, resulting in either an equilibrium point, or a tariff cycle (Johnson,

1953). Another way of modeling the tariff war was proposed by Tower who treated retaliation as a

Stackelberg leader/follower situation which country one perceived itself as a leader which country two as

a follower (Tower, 1976). In our simulation, we followed a similar setting as Johnson’s Cournot game.

The difference was that in each round only one player moved based on the action of another player from

previous round. So player set his best respond (optimal tariff) at each round considering another player’s

best respond from last round. If the best responses converges, then the result should be the same as that in

a Cournot behavioral game.

8

10

12

14

16

18

20

22

24

26

28

1 7 5 1 9 5 2 1 5 2 3 5 2 5 5 2 7 5 2 9 5

US TAX REVENUE

US Tax Revenue

17

Given the initial setting, we let US move first. So US moves in all odd periods and UK moves in

all even periods. Each one plays ten rounds so we have twenty rounds in total. The following graph plots

each player’s optimal tariff in each round. We can see that after 8 rounds, both of their optimal tariffs

converges to 0.17. Their utilities converges to 272.6. The joint utility under such tariff war is 272.6 * 2 =

545.2. Comparing to previous scenarios, it is lower than those under free trade (550) and unilateral

optimal tariff (547.82). Such result is similar as duopoly price war under collusion and Cournot game.

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

OPTIMAL TARIFF

US Optimal Beer Tariff UK Optimal Auto Tariff

18

In the above tariff war, the convergent optimal tariffs and utilities are equal. This is due to the

symmetric setting of our model. Next, we simulate a tariff war using unsymmetric setting of countries. In

the following simulation, we set Beta('auto','US')=0.7; Beta('beer','US')=0.3. So US has a higher share of

expenditure on auto than beer. On the other hand, UK maintain its betas equaling to 0.5 for both auto and

beer.

In the following set of graphs, the left side shows the results which US moves first, and those on

the right side represent the result which UK moves first. It turns out that the convergent optimal tariffs

and utilities are independent of who starts first. In other words, in terms of the end value, the first mover

does not have anvantage. The unsymmetric settings of country generate unsymmetirc convergent values.

In this case, US has higher end value of utility but lower value of opimal import tariff.

269

270

271

272

273

274

275

276

277

278

279

0 1 2 3 4 5 6 7 8 9 1 0

UTILITY


19

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 1 0

OPTIMAL TARIFF

US Optimal Beer Tariff

UK Optimal Auto Tariff

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 1 0

OPTIMAL TARIFF

US Optimal Beer Tariff

UK Optimal Auto Tariff

265

270

275

280

285

290

295

300

305

0 2 4 6 8 1 0

UTILITY


265

270

275

280

285

290

295

300

305

0 2 4 6 8 1 0

UTILITY


20

Reference Johnson, H. G. (1953). Optimum Tariffs and Retaliation. The Review of Economic Studies, 21(2), 142–

153. doi:10.2307/2296006

Tower, E. (1976). The Optimum Tariff, Retaliation and Autarky. Eastern Economic Journal, 3(2), 72–75.

21

Appendix * Adapted from "two country version of the HOS model with trade

taxes", as described in Chapter 19 of

* Gilbert and Tower: Introduction to Numerical Simulation for Trade

Theory and Policy.

*Setting of the model

* Define the indexes for the problem

SET I Goods /Auto,Beer/;

SET J Factors /K,L/;

SET D Countries /US,UK/;

ALIAS (J, JJ);

* Create names for parameters

PARAMETERS

ALPHA(D) Shift parameter in utility

BETA(I,D) Share parameter in utility

PO(I,D) Initial domestic prices

PWO(I) Initial world prices

XO(I,D) Initial trade

UO(D) Initial utility levels

CO(I,D) Initial consumption levels

GAMMA(I,D) Shift parameters in production

DELTA(J,I,D) Share parameters in production

RHO(I,D) Elasticity parameters in production

FBAR(J,D) Factor Endowments

QO(I,D) Initial output levels

RO(J,D) Initial factor prices

FO(J,I,D) Initial factor use levels

GDPO(D) Initial gross domestic product;

* Assign values to the parameters

PWO(I)=1;

PO(I,D)=1;

RO(J,D)=1;

PARAMETER QO(I,D) Initial output levels /

Auto.US 200

Auto.UK 75

Beer.US 75

Beer.UK 200 /;

CO(I,D)=137.5;

XO(I,D)=QO(I,D)-CO(I,D);

PARAMETER FO(J,I,D) Initial factor use levels /

22

K.Auto.US 160

K.Auto.UK 60

K.Beer.US 15

K.Beer.UK 40

L.Auto.US 40

L.Auto.UK 15

L.Beer.US 60

L.Beer.UK 160 /;

FBAR(J,D)=SUM(I, FO(J,I,D));

GDPO(D)=SUM(I, PO(I,D)*QO(I,D));

RHO(I,D)=0.1;

DELTA(J,I,D)=(RO(J,D)/FO(J,I,D)**(RHO(I,D)-1))/(SUM(JJ,

RO(JJ,D)/FO(JJ,I,D)**(RHO(I,D)-1)));

GAMMA(I,D)=QO(I,D)/(SUM(J,

DELTA(J,I,D)*FO(J,I,D)**RHO(I,D)))**(1/RHO(I,D));

UO(D)=GDPO(D);

BETA(I,D)=CO(I,D)/GDPO(D);

ALPHA(D)=UO(D)/PROD(I, CO(I,D)**BETA(I,D));

* Create names for variables

VARIABLES

U(D) Utility indices

T(I,D)

P(I,D) Prices

C(I,D) Consumption levels

Q(I,D) Output levels

R(J,D) Factor prices

F(J,I,D) Factor use levels

PW(I) World prices

X(I,D) Trade levels

GDP(D) Gross domestic products

OBJ Objective;

* Assign initial values to variables, and set lower bounds

U.L(D)=UO(D);

P.L(I,D)=PO(I,D);

C.L(I,D)=CO(I,D);

Q.L(I,D)=QO(I,D);

R.L(J,D)=RO(J,D);

F.L(J,I,D)=FO(J,I,D);

PW.L(I)=PWO(I);

X.L(I,D)=XO(I,D);

GDP.L(D)=GDPO(D);

OBJ.L=SUM(D, UO(D));

P.LO(I,D)=0;

C.LO(I,D)=0;

Q.LO(I,D)=0;

R.LO(J,D)=0;

F.LO(J,I,D)=0;

23

GDP.LO(D)=0;

PW.LO(I)=0;

* Fix the value of one price to serve as a numeraire

PW.FX('Auto')=1;

* Create names for equations

EQUATIONS

UTILITY(D) Utility functions

DEMAND(I,D) Demand functions

PRODUCTION(I,D) Production functions

RESOURCE(J,D) Resource constraints

FDEMAND(J,I,D) Factor demand functions

INCOME(D) Gross domestic products

MAT_BAL(I,D) Open economy material balance

INT_CLEAR(I) International market clearing

ARBITRAGE(I,D) International price arbitrage

OBJECTIVE_SUM Objective function

OBJECTIVE_US Objective function

OBJECTIVE_UK Objective function;

* Assign the expressions to the equation names

UTILITY(D)..U(D)=E=ALPHA(D)*PROD(I, C(I,D)**BETA(I,D));

DEMAND(I,D)..C(I,D)=E=BETA(I,D)*GDP(D)/P(I,D);

MAT_BAL(I,D)..X(I,D)=E=Q(I,D)-C(I,D);

INT_CLEAR(I)..SUM(D, X(I,D))=E=0;

ARBITRAGE(I,D)..P(I,D)=E=(1+T(I,D))*PW(I);

PRODUCTION(I,D)..Q(I,D)=E=GAMMA(I,D)*SUM(J,

DELTA(J,I,D)*F(J,I,D)**RHO(I,D))**(1/RHO(I,D));

RESOURCE(J,D)..FBAR(J,D)=E=SUM(I, F(J,I,D));

FDEMAND(J,I,D)..R(J,D)=E=P(I,D)*Q(I,D)*SUM(JJ,

DELTA(JJ,I,D)*F(JJ,I,D)**RHO(I,D))**(-

1)*DELTA(J,I,D)*F(J,I,D)**(RHO(I,D)-1);

INCOME(D)..GDP(D)=E=SUM(I, P(I,D)*Q(I,D))-SUM(I, PW(I)*T(I,D)*X(I,D));

OBJECTIVE_US..OBJ=E=U('US');

OBJECTIVE_UK..OBJ=E=U('UK');

OBJECTIVE_SUM..OBJ=E=SUM(D,U(D));

MODEL TRADE / UTILITY, DEMAND, MAT_BAL, INT_CLEAR, ARBITRAGE,

PRODUCTION, RESOURCE, FDEMAND, INCOME, OBJECTIVE_SUM /;

MODEL TRADE_US / UTILITY, DEMAND, MAT_BAL, INT_CLEAR, ARBITRAGE,

PRODUCTION, RESOURCE, FDEMAND, INCOME, OBJECTIVE_US /;

MODEL TRADE_UK / UTILITY, DEMAND, MAT_BAL, INT_CLEAR, ARBITRAGE,

PRODUCTION, RESOURCE, FDEMAND, INCOME, OBJECTIVE_UK /;

*1.solve the model of two countries with no tariff


24

T.FX(I,D)=0;

Solve TRADE USING NLP MAXIMIZING OBJ;

execute_unload"results.GDX" U T P C Q R F PW X GDP

execute'gdxxrw.exe results.GDX var=U rng=s1!D1:E2'

execute'gdxxrw.exe results.GDX var=T rng=s1!C3:E5'

execute'gdxxrw.exe results.GDX var=P rng=s1!C6:E8'

execute'gdxxrw.exe results.GDX var=C rng=s1!C9:E11'

execute'gdxxrw.exe results.GDX var=Q rng=s1!C12:E14'

execute'gdxxrw.exe results.GDX var=R rng=s1!C15:E17'

execute'gdxxrw.exe results.GDX var=F rng=s1!B18:G22'

execute'gdxxrw.exe results.GDX var=PW rng=s1!D23:E24'

execute'gdxxrw.exe results.GDX var=X rng=s1!C25:E27'

execute'gdxxrw.exe results.GDX var=GDP rng=s1!D28:E29';

*2. US sets import tariff to maximize US utility, given no

tax in UK


T.FX(I,'UK')=0;

T.FX('Auto','US')=0;

T.LO('beer','US')=-INF;

T.UP('beer','US')=INF;

Solve TRADE_US USING NLP MAXIMIZING OBJ;












*3. US sets export tax/subsidy to maximize US utility,

given no tax in UK


T.FX(I,'UK')=0;

T.FX('beer','US')=0;

T.LO('auto','US')=-INF;

T.UP('auto','US')=INF;


25












*4. US sets both import and export taxes at 10%, given no

tariff of UK (Lerner Symmetry Theorem).


T.FX(I,'UK')=0;

T.FX('beer','US')=0.1;

T.FX('auto','US')=0.1;













*5.Demonstration of Factor equalization theorem

*assumptions hold, change the relative K/L ratio

RHO(I,D)=0.1;

T.FX(I,D)=0;

PARAMETER FBAR(J,D)/

K.US = 100

K.UK = 100

L.US = 100

L.UK = 175/;


26












*assumptions fail (factor intensity reversal) and change the relative

K/L ratio

RHO(I,D)=0.00001;

T.FX(I,D)=0;

PARAMETER FBAR(J,D)/

K.US = 9999

K.UK = 100

L.US = 10

L.UK = 200/;













*6. Plot US Import Tariff vs. U, C, X and Q, given no tax

in UK


T.FX(I,D)=0;

SET ITERR /1*99/;

PARAMETER

Us(ITERR)

27

Ts(ITERR)

PAs(ITERR)

PBs(ITERR)

CAs(ITERR)

CBs(ITERR)

QAs(ITERR)

QBs(ITERR)

RKs(ITERR)

RLs(ITERR)

XAss(ITERR)

XBss(ITERR)

;

LOOP(ITERR,

SOLVE TRADE_US USING NLP MAXIMIZING OBJ;

Us(ITERR)=U.L('US');

Ts(ITERR)=T.L('Beer','US');

PAs(ITERR)=P.L('Auto','US');

PBs(ITERR)=P.L('Beer','US');

CAs(ITERR)=C.L('Auto','US');

CBs(ITERR)=C.L('Beer','US');

QAs(ITERR)=Q.L('Auto','US');

QBs(ITERR)=Q.L('Beer','US');

RKs(ITERR)=R.L('K','US');

RLs(ITERR)=R.L('L','US');

XAss(ITERR)=X.L('Auto','US');

XBss(ITERR)=X.L('Beer','US');

T.FX('Beer','US')=T.L('Beer','US')+0.05;

);

execute_unload"s5plot.GDX" Us Ts PAs PBs CAs CBs QAs QBs RKs RLs XAss

XBss

execute'gdxxrw.exe s5plot.GDX par=Ts rng=s5!B2:CX2'

execute'gdxxrw.exe s5plot.GDX par=Us rng=s5!B4:CX4'

execute'gdxxrw.exe s5plot.GDX par=PAs rng=s5!B6:CX6'

execute'gdxxrw.exe s5plot.GDX par=PBs rng=s5!B8:CX8'

execute'gdxxrw.exe s5plot.GDX par=CAs rng=s5!B10:CX10'

execute'gdxxrw.exe s5plot.GDX par=CBs rng=s5!B12:CX12'

execute'gdxxrw.exe s5plot.GDX par=QAs rng=s5!B14:CX14'

execute'gdxxrw.exe s5plot.GDX par=QBs rng=s5!B16:CX16'

execute'gdxxrw.exe s5plot.GDX par=RKs rng=s5!B18:CX18'

execute'gdxxrw.exe s5plot.GDX par=RLs rng=s5!B20:CX20'

execute'gdxxrw.exe s5plot.GDX par=XAss rng=s5!B22:CX22'

execute'gdxxrw.exe s5plot.GDX par=XBss rng=s5!B24:CX24';

*7a.Sensitivity Analysis on Beta('US')


T.FX(I,'UK')=0;




28

Beta('auto','US')=0.3;

Beta('beer','US')=0.7;

SET ITERRR /1*69/;

PARAMETER

Usa(ITERRR)

Usaa(ITERRR)

Tsa(ITERRR)

PAsa(ITERRR)

PBsa(ITERRR)

CAsa(ITERRR)

CBsa(ITERRR)

QAsa(ITERRR)

QBsa(ITERRR)

RKsa(ITERRR)

RLsa(ITERRR)

XAssa(ITERRR)

XBssa(ITERRR)

Betasa(ITERRR)

;

LOOP(ITERRR,


Usa(ITERRR)=U.L('US');

Usaa(ITERRR)=U.L('UK');

Tsa(ITERRR)=T.L('Beer','US');

PAsa(ITERRR)=P.L('Auto','US');

PBsa(ITERRR)=P.L('Beer','US');

CAsa(ITERRR)=C.L('Auto','US');

CBsa(ITERRR)=C.L('Beer','US');

QAsa(ITERRR)=Q.L('Auto','US');

QBsa(ITERRR)=Q.L('Beer','US');

RKsa(ITERRR)=R.L('K','US');

RLsa(ITERRR)=R.L('L','US');

XAssa(ITERRR)=X.L('Auto','US');

XBssa(ITERRR)=X.L('Beer','US');

BETAsa(ITERRR)=Beta('auto','US');

Beta('auto','US')=Beta('auto','US')+0.01;

Beta('beer','US')=Beta('beer','US')-0.01;

);

execute_unload"s7aplot.GDX" Betasa Usa Usaa Tsa PAsa PBsa CAsa CBsa

QAsa QBsa RKsa RLsa XAssa XBssa

execute'gdxxrw.exe s7aplot.GDX par=Betasa rng=s5!B2:CX2'

execute'gdxxrw.exe s7aplot.GDX par=Usa rng=s5!B4:CX4'

execute'gdxxrw.exe s7aplot.GDX par=PAsa rng=s5!B6:CX6'

execute'gdxxrw.exe s7aplot.GDX par=PBsa rng=s5!B8:CX8'

execute'gdxxrw.exe s7aplot.GDX par=CAsa rng=s5!B10:CX10'

execute'gdxxrw.exe s7aplot.GDX par=CBsa rng=s5!B12:CX12'

execute'gdxxrw.exe s7aplot.GDX par=QAsa rng=s5!B14:CX14'

execute'gdxxrw.exe s7aplot.GDX par=QBsa rng=s5!B16:CX16'

execute'gdxxrw.exe s7aplot.GDX par=RKsa rng=s5!B18:CX18'

execute'gdxxrw.exe s7aplot.GDX par=RLsa rng=s5!B20:CX20'

29

execute'gdxxrw.exe s7aplot.GDX par=XAssa rng=s5!B22:CX22'

execute'gdxxrw.exe s7aplot.GDX par=XBssa rng=s5!B24:CX24'

execute'gdxxrw.exe s7aplot.GDX par=Tsa rng=s5!B26:CX26';

execute'gdxxrw.exe s7aplot.GDX par=Usaa rng=s5!B28:CX28';

*7b.Sensitivity Analysis on US Capital


T.FX(I,'UK')=0;




SET ITERRR /1*100/;

PARAMETER

Usb(ITERRR)

Usbb(ITERRR)

Tsb(ITERRR)

PAsb(ITERRR)

PBsb(ITERRR)

CAsb(ITERRR)

CBsb(ITERRR)

QAsb(ITERRR)

QBsb(ITERRR)

RKsb(ITERRR)

RLsb(ITERRR)

XAssb(ITERRR)

XBssb(ITERRR)

Betasb(ITERRR)

Rsb(ITERRR)

;

LOOP(ITERRR,


Usb(ITERRR)=U.L('US');

Usbb(ITERRR)=U.L('UK');

Tsb(ITERRR)=T.L('Beer','US');

PAsb(ITERRR)=P.L('Auto','US');

PBsb(ITERRR)=P.L('Beer','US');

CAsb(ITERRR)=C.L('Auto','US');

CBsb(ITERRR)=C.L('Beer','US');

QAsb(ITERRR)=Q.L('Auto','US');

QBsb(ITERRR)=Q.L('Beer','US');

RKsb(ITERRR)=R.L('K','US');

RLsb(ITERRR)=R.L('L','US');

XAssb(ITERRR)=X.L('Auto','US');

XBssb(ITERRR)=X.L('Beer','US');

BETAsb(ITERRR)=Beta('auto','US');

Rsb(ITERRR)=T.L('Beer','US')*P.L('Beer','UK')*X.L('Beer','UK');

FBAR('K','US')=FBAR('K','US')+2;

);

30

execute_unload"s7bplot.GDX" Usb Usbb Rsb Tsb PAsb PBsb CAsb CBsb QAsb

QBsb RKsb RLsb XAssb XBssb

execute'gdxxrw.exe s7bplot.GDX par=Usbb rng=s5!B2:CW2'

execute'gdxxrw.exe s7bplot.GDX par=Usb rng=s5!B4:CW4'

execute'gdxxrw.exe s7bplot.GDX par=PAsb rng=s5!B6:CW6'

execute'gdxxrw.exe s7bplot.GDX par=PBsb rng=s5!B8:CW8'

execute'gdxxrw.exe s7bplot.GDX par=CAsb rng=s5!B10:CW10'

execute'gdxxrw.exe s7bplot.GDX par=CBsb rng=s5!B12:CW12'

execute'gdxxrw.exe s7bplot.GDX par=QAsb rng=s5!B14:CW14'

execute'gdxxrw.exe s7bplot.GDX par=QBsb rng=s5!B16:CW16'

execute'gdxxrw.exe s7bplot.GDX par=RKsb rng=s5!B18:CW18'

execute'gdxxrw.exe s7bplot.GDX par=RLsb rng=s5!B20:CW20'

execute'gdxxrw.exe s7bplot.GDX par=XAssb rng=s5!B22:CW22'

execute'gdxxrw.exe s7bplot.GDX par=XBssb rng=s5!B24:CW24'

execute'gdxxrw.exe s7bplot.GDX par=Tsb rng=s5!B26:CW26';

execute'gdxxrw.exe s7bplot.GDX par=Rsb rng=s5!B28:CW28';

*8. Trade War

*US starts first, set the optimal tariff, given UK tax fixed.

*UK then sets its optimal tariff, given US tariff rate from previous

step.

*US moves...

*100 rounds.

BETA('Auto','US')=0.7;

BETA('Beer','US')=0.3;

T.FX(I,D)=0;

SET ITER /1*10/;

PARAMETER

UUSs(ITER)

UUKs(ITER)

TUSBs(ITER)

TUKAs(ITER)

CAUSs(ITER)

CBUSs(ITER)

CAUKs(ITER)

CBUKs(ITER)

XAs(ITER)

XBs(ITER)

;

Loop(ITER,

T.FX('Auto','UK')=T.L('Auto','UK');

T.LO('Beer','US')=-INF;

T.UP('Beer','US')=INF;


31

UUSs(ITER)=U.L('US');

UUKs(ITER)=U.L('UK');

TUSBs(ITER)=T.L('Beer','US');

TUKAs(ITER)=T.L('Auto','UK');

CAUSs(ITER)=C.L('Auto','US');

CBUSs(ITER)=C.L('Beer','US');

CAUKs(ITER)=C.L('Auto','UK');

CBUKs(ITER)=C.L('Beer','UK');

XAs(ITER)=X.L('Auto','US');

XBs(ITER)=X.L('Beer','UK');

T.FX('Beer','US')=T.L('Beer','US');

T.LO('Auto','UK')=-INF;

T.UP('Auto','UK')=INF;

SOLVE TRADE_UK USING NLP MAXIMIZING OBJ;

);

execute_unload"S8.GDX" UUSs UUKs TUSBs TUKAs CAUSs CBUSs CAUKs CBUKs

XAs XBs

execute'gdxxrw.exe S8.GDX par=TUSBs rng=US!B2:GT2'

execute'gdxxrw.exe S8.GDX par=TUKAs rng=US!B4:GT4'

execute'gdxxrw.exe S8.GDX par=UUSs rng=US!B6:GT6'

execute'gdxxrw.exe S8.GDX par=UUKs rng=US!B8:GT8'

execute'gdxxrw.exe S8.GDX par=CAUSs rng=US!B10:GT10'

execute'gdxxrw.exe S8.GDX par=CBUSs rng=US!B12:GT12'

execute'gdxxrw.exe S8.GDX par=CAUKs rng=US!B14:GT14'

execute'gdxxrw.exe S8.GDX par=CBUKs rng=US!B16:GT16'

execute'gdxxrw.exe S8.GDX par=XAs rng=US!B18:GT18'

execute'gdxxrw.exe S8.GDX par=XBs rng=US!B20:GT20';

T.FX(I,D)=0;

Loop(ITER,

T.FX('Auto','UK')=T.L('Auto','UK');

T.LO('Beer','US')=-INF;

T.UP('Beer','US')=INF;


T.FX('Beer','US')=T.L('Beer','US');

T.LO('Auto','UK')=-INF;

T.UP('Auto','UK')=INF;

SOLVE TRADE_UK USING NLP MAXIMIZING OBJ;

UUSs(ITER)=U.L('US');

UUKs(ITER)=U.L('UK');

TUSBs(ITER)=T.L('Beer','US');

TUKAs(ITER)=T.L('Auto','UK');

32

CAUSs(ITER)=C.L('Auto','US');

CBUSs(ITER)=C.L('Beer','US');

CAUKs(ITER)=C.L('Auto','UK');

CBUKs(ITER)=C.L('Beer','UK');

XAs(ITER)=X.L('Auto','US');

XBs(ITER)=X.L('Beer','UK');

);

execute_unload"S8.GDX" UUSs UUKs TUSBs TUKAs CAUSs CBUSs CAUKs CBUKs

XAs XBs

execute'gdxxrw.exe S8.GDX par=TUSBs rng=UK!B2:GT2'

execute'gdxxrw.exe S8.GDX par=TUKAs rng=UK!B4:GT4'

execute'gdxxrw.exe S8.GDX par=UUSs rng=UK!B6:GT6'

execute'gdxxrw.exe S8.GDX par=UUKs rng=UK!B8:GT8'

execute'gdxxrw.exe S8.GDX par=CAUSs rng=UK!B10:GT10'

execute'gdxxrw.exe S8.GDX par=CBUSs rng=UK!B12:GT12'

execute'gdxxrw.exe S8.GDX par=CAUKs rng=UK!B14:GT14'

execute'gdxxrw.exe S8.GDX par=CBUKs rng=UK!B16:GT16'

execute'gdxxrw.exe S8.GDX par=XAs rng=UK!B18:GT18'

execute'gdxxrw.exe S8.GDX par=XBs rng=UK!B20:GT20'

Optimum Tariff and Finite Sequential Trade...

Documents

Transcript of Optimum Tariff and Finite Sequential Trade...