comparative advantage and optimal trade taxes - … · comparative advantage and optimal trade...

Comparative Advantageand Optimal Trade Taxes

Arnaud Costinot (MIT), Dave Donaldson (MIT),Jonathan Vogel (Columbia) and Iván Werning (MIT)

June 2014

Motivation

• Two central questions...

1. Why do nations trade?

2. How should they conduct trade policy?

• Theory of comparative advantage Influential answer to #1 v Virtually no impact on #2

This Paper

• Take canonical Ricardian model

• simplest and oldest theory of CA

• new workhorse model for theoretical and quantitative work

• Explore relationship... CA Optimal Trade Taxes

Main Result

• Optimal trade taxes:

1. uniform across imported goods

2. monotone in CA across exported goods

Main Result

• Examples:

export taxesincreasing in CA

export subsidiesdecreasing in CA

+

+

zero import tariff

Positive import tariff

Simple Economics

Simple Economics

• More room to manipulate prices in comparative advantage sectors

Simple Economics


• New perspective on targeted industrial policy

Simple Economics



• larger subsidies for less competitive sectors not from desire to expand output ...

Simple Economics



• larger subsidies for less competitive sectors not from desire to expand output ...

• ... but greater constraints to contract exports to exploit monopoly power

Two Applications

• Agriculture and Manufacturing examples

• GT under optimal trade taxes are 20% and 33% larger than under no taxes

• GT under under optimal uniform tariff are only 9% larger than under no taxes

• Micro-level heterogeneity matters for design and gains from optimal trade policy

Related Literature

• Optimal Taxes in an Open Economy:

• General results: Dixit (85), Bond (90)

• Ricardo: Itoh Kiyono (87), Opp (09)

• Lagrangian Methods:

• Lagrangian methods in infinite dimensional space: AWA (06), Amador Bagwell (13)

• Cell-problems: Everett (63), CLW (13)

Roadmap

• Basic Environment

• Optimal Allocation

• Optimal Trade Taxes

• Applications

Basic Environment

A Ricardian Economy• Two countries: Home and Foreign

• Labor endowments: and

• CES utility over continuum of goods:

• Constant unit labor requirements: and

• Home sets trade taxes and lump-sum transfer

• Foreign is passive

U ⌘Z

iui(ci)di

ui(ci) ⌘ �i

⇣c1�1/�

i � 1⌘.

(1� 1/�)

L L⇤

ai a⇤i

t ⌘ (ti) T

Competitive Equilibrium

Competitive Equilibriumc 2 argmaxc̃�0

⇢Z

iui(c̃i)di

��Z

ipi (1 + ti) c̃idi wL+ T

�


⇢Z

iui(c̃i)di

��Z


�

qi 2 argmaxq̃i�0 {pi (1 + ti) q̃i � waiq̃i}


⇢Z

iui(c̃i)di

��Z


�


T =

Z

ipiti (ci � qi) di


⇢Z

iui(c̃i)di

��Z


�


T =

Z


c⇤ 2 argmaxc̃�0

⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�


⇢Z

iui(c̃i)di

��Z


�


T =

Z



⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�

q⇤i 2 argmaxq̃i�0 {piq̃i � w⇤a⇤i q̃i}


⇢Z

iui(c̃i)di

��Z


�


T =

Z



⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�

q⇤i 2 argmaxq̃i�0 {piq̃i � w⇤a⇤i q̃i}ci + c⇤i = qi + q⇤i ,


⇢Z

iui(c̃i)di

��Z


�


T =

Z



⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�

q⇤i 2 argmaxq̃i�0 {piq̃i � w⇤a⇤i q̃i}ci + c⇤i = qi + q⇤i ,Z

iaiqidi = L,


⇢Z

iui(c̃i)di

��Z


�


T =

Z



⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�

q⇤i 2 argmaxq̃i�0 {piq̃i � w⇤a⇤i q̃i}ci + c⇤i = qi + q⇤i ,Z

iaiqidi = L,

Z

ia⇤i q

⇤i di = L⇤.

Government Problem

Government Problem

c 2 argmaxc̃�0

⇢Z

iui(c̃i)di

��Z


�


T =

Z



⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�

q⇤i 2 argmaxq̃i�0 {piq̃i � w⇤a⇤i q̃i}ci + c⇤i = qi + q⇤iZ

iaiqidi = L,

Z

ia⇤i q

⇤i di = L⇤.

Government Problem

c 2 argmaxc̃�0

⇢Z

iui(c̃i)di

��Z


�


T =

Z



⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�


iaiqidi = L,

Z

ia⇤i q

⇤i di = L⇤.

s.t.t, T, w,w⇤, p, c, c⇤, q, q⇤

max

U(c)

Optimal Allocation

Let us Relax

• Primal approach (Baldwin 48, Dixit 85):

c q

No taxes, no competitive markets at home

Domestic government directly controls domestic consumption, , and output,

Planning Problem

s.t.c 2 argmaxc̃�0

⇢Z

iui(c̃i)di

��Z


�


T =

Z



⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�


iaiqidi = L,

Z

ia⇤i q

⇤i di = L⇤.

t, T, w,w⇤, p, c, c⇤, q, q⇤max

U(c)

Planning Problem

s.t.c⇤ 2 argmaxc̃�0

⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�


iaiqidi = L,

Z

ia⇤i q

⇤i di = L⇤.

t, T, w,w⇤, p, c, c⇤, q, q⇤max

U(c)

Planning Problem


⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�


iaiqidi = L,

Z

ia⇤i q

⇤i di = L⇤.

w⇤, p, c, c⇤, q, q⇤max

U(c)

• Convenient to focus on 3 key controls:

• Equilibrium abroad requires...

pi (mi, w⇤) ⌘ min {u⇤0

i (�mi) , w⇤a⇤i } ,

q⇤i (mi, w⇤) ⌘ max {mi + d⇤i (w

⇤a⇤i ), 0}

Planning Problem

Planning Problem


⇢Z

iu⇤i (c̃i)di

��Z

ipic̃idi w⇤L⇤

�


iaiqidi = L,

Z

ia⇤i q

⇤i di = L⇤.

w⇤, p, c, c⇤, q, q⇤max

U(c)

Planning Problem

s.t.w⇤, p, c, c⇤, q, q⇤max

U(c)

Planning Problem


U(c)

Z

iaiqidi L,

Planning Problem


U(c)

Z

iaiqidi L,

Z

ia⇤i q

⇤i (mi, w

⇤) di L⇤,

Planning Problem


U(c)

Z

iaiqidi L,

Z

ia⇤i q

⇤i (mi, w

⇤) di L⇤,Z

ipi(mi, w

⇤)midi 0

Planning Problem

s.t.

max

U(c)

Z

iaiqidi L,

Z

ia⇤i q

⇤i (mi, w

⇤) di L⇤,Z

ipi(mi, w

⇤)midi 0

w⇤,m, q

Planning Problem

s.t.

max

Z

iaiqidi L,

Z

ia⇤i q

⇤i (mi, w

⇤) di L⇤,Z

ipi(mi, w

⇤)midi 0

U(m+ q)w⇤,m, q

Three Steps1. Decompose

(i) inner problem

(ii) outer problem

2. Concavity of inner problem Lagrangian Theorems (Luenberger 69)

3. Additive separability implies... (Everett 63)one infinite-dimensional problem many low-dimensional problems

w⇤

Inner Problem

s.t.

max

Z

iaiqidi L,

Z

ia⇤i q

⇤i (mi, w

⇤) di L⇤,Z

ipi(mi, w

⇤)midi 0

U(m+ q)w⇤,m, q

Inner Problem

s.t.

max

Z

iaiqidi L,

Z

ia⇤i q

⇤i (mi, w

⇤) di L⇤,Z

ipi(mi, w

⇤)midi 0

U(m+ q)m, q

Lagrangian

for some and

Lagrangian Theorem• solves inner problem iff

(�,�⇤, µ)

� � 0,

Z

iaiq

0i di L, with complementary slackness,

�⇤ � 0,

Z

ia⇤i q

⇤i

�m0

i , w⇤� di L⇤

, with complementary slackness,

µ � 0,

Z

ipi(mi, w

⇤)m0

i di 0, with complementary slackness.

�m0, q0

�

max

m,qL (m, q,�,�⇤, µ;w⇤

)

for some and

Cell Structure

� � 0,

Z

iaiq

0i di L, with complementary slackness,

�⇤ � 0,

Z

ia⇤i q

⇤i

�m0

i , w⇤� di L⇤

, with complementary slackness,

µ � 0,

Z

ipi(mi, w

⇤)m0

i di 0, with complementary slackness.

• solves inner problem iff solves�m0

i , q0i

��m0, q0

�

(�,�⇤, µ)

max

mi,qiLi (mi, qi,�,�

⇤, µ;w⇤)

High-School Math:Optimal Output

High-School Math:Optimal Output

mi

qi, q⇤i

MIi 0 MII

i

fdsfdasfda

1

High-School Math:Optimal Net Imports

High-School Math:Optimal Net Imports

mi

Li

mIi MI

i 0 MIIi

(a) ai/a⇤i < AI .

mi

Li

MIi 0 MII

i

(b) ai/a⇤i 2 [AI , AII).

mi

Li

MIi 0 MII

i

(c) ai/a⇤i = AII .

mi

Li

MIi 0 MII

i mII Ii

(d) ai/a⇤i > AII .

Figure 1: Optimal net imports.

fdsfdasfdasfdsafdsafdsafdsfdsafdsfdsfdsfdsfdsfdsfdsffdsfdsf

1

Optimal Trade Taxes

Wedges

• Wedges at planning problem’s solution:

• Previous analysis implies:

⌧0i ⌘u0i

�c0i�

p0i� 1

⌧0i =

8>><

>>:

�⇤�1�⇤ µ0 � 1, if ai

a⇤i< AI ⌘ �⇤�1

�⇤µ0w0⇤

�0 ;�0aiw0⇤a⇤

i� 1, if AI < ai

a⇤i AII ⌘ µ0w0⇤+�0⇤

�0 ;�0⇤

w0⇤ + µ0 � 1, if aia⇤i> AII .

• Any solution to Home's planning problem can be implemented by

• Conversely, if solves the domestic's government problem, then the associated allocation and prices must solve Home’s planning problem and satisfy:

Optimal Trade Taxes

t0 = ⌧0

t0i =u0i

�c0i�

✓p0i� 1

t0

Optimal Trade Taxes

Intuition

• When , Home has incentives to charge constant monopoly markup

• When , there is limit pricing: foreign firms are exactly indifferent between producing and not producing those goods

• When , uniform tariff is optimal: Home cannot manipulate relative prices

ai/a⇤i < AI

ai/a⇤i 2

⇥AI , AII

⇤

ai/a⇤i > AII

Industrial Policy Revisited


• At the optimal policy, governments protects a subset of less competitive industries

• but targeted/non-uniform subsidies do not stem from a greater desire to expand production...

• ... they reflect tighter constraints on ability to exploit monopoly power by contracting exports


• At the optimal policy, governments protects a subset of less competitive industries

• but targeted/non-uniform subsidies do not stem from a greater desire to expand production...

• ... they reflect tighter constraints on ability to exploit monopoly power by contracting exports

• Countries have more room to manipulate world prices in their comparative-advantage sectors

Robustness

• Similar qualitative results hold in more general environments:

• Iceberg trade costs

• Separable, but non-CES utility

• Additional considerations:

• Trade costs imply that zero imports are optimal for some goods at solution of Home’s planning problem

• Non-CES utility leads to variable markups for goods with strongest CA

Applications

Agricultural Example

• Home = U.S. Foreign = R.O.W.

• Each good corresponds to 1 of 39 crops

• Land is the only factor of production

• Productivity from FAO’s GAEZ project

• Land endowments match acreage devoted to 39 crops in U.S. and R.O.W.

• Symmetric CES utility with σ=2.9 as in BW (06)

Optimal Trade Taxes

Optimal Trade Taxes

0 0.4 0.8 1.2 1.6 2-60%

-40%

-20%

0

ai/a⇤i

t0 i

0 0.4 0.8 1.2 1.6 2-60%

-40%

-20%

0

ai/a⇤i

t0 i

Figure 3: Optimal trade taxes for the agricultural case. The left panel assumes no tradecosts, d = 0. The right panel assumes trade costs, d = 1.72.

crops i as a function of comparative advantage, ai/a⇤i , in the calibrated examples withouttrade costs, d = 1, and with trade costs, d = 1.72, respectively.12 The region betweenthe two vertical lines in the right panel corresponds to goods that are not traded at thesolution of Home’s planning problem.

As discussed in Section 4.2, the overall level of taxes is indeterminate. Figure 3 fo-cuses on a normalization with zero import tariffs. In both cases, the maximum export taxis close to the optimal monopoly markup that a domestic firm would have charged on theforeign market, s/ (s � 1)� 1 ' 52.6%. The only difference between the two markupscomes from the fact that the domestic government internalizes the effect that the net im-ports of each good have on the foreign wage. Specifically, if the Lagrange multiplier onthe foreign resource constraint, l0⇤, was equal to zero, then the maximum export tax,which is equal AI/AII � 1, would simplify into the firm-level markup, s/ (s � 1)� 1. Inother words, such general equilibrium considerations appear to have small effects on thedesign of optimal trade taxes for goods in which the U.S. comparative advantage is thestrongest. In light of the discussion in Section 4.3, these quantitative results suggest thatif domestic firms were to act as monopolists rather than take prices as given, then the do-mestic government could get close to the optimal allocation by only using consumption

12We compute optimal trade taxes, throughout this and the next subsection, by performing a grid searchover the foreign wage w⇤ so as to maximize V(w⇤). Since Foreign cannot be worse off under trade thanunder autarky—whatever world prices may be, there are gains from trade—and cannot be better off thanunder free trade—since free trade is a Pareto optimum, Home would have to be worse off—we restrictour grid search to values of the foreign wage between those that would prevail in the autarky and freetrade equilibria. Recall that we have normalized prices so that the Lagrange multiplier associated with theforeign budget constraint is equal to one. Thus w⇤ is the real wage abroad.

26

Gains from Trade

Gains from Trade

No Trade CostsNo Trade Costs Trade CostsTrade Costs

U.S. R.O.W. U.S. R.O.W.

Laissez-Faire 39.15% 3.02% 5.02% 0.25%

Uniform Tariff 42.60% 1.41% 5.44% 0.16%

Optimal Taxes 46.92% 0.12% 5.71% 0.04%

Manufacturing Example• Home=U.S. and Foreign=R.O.W.

• 400 goods. Labor is the only factor of production

• Labor endowments set to match population in U.S. and R.O.W

• Productivity is distributed Fréchet:

• θ=5 set to match average trade elasticity in HM (13).

• T and T* set to match U.S. share of world GDP.

• Symmetric CES utility with σ=2.5 as in BW (06)

ai =

✓i

T

◆ 1✓

and a⇤i =

✓1� i

T ⇤

◆ 1✓

Optimal Trade Taxes

Optimal Trade Taxes

0 0.2 0.4 0.6

-60%

-40%

-20%

0

ai/a⇤i

t0 i

0 0.2 0.4 0.6

-60%

-40%

-20%

0

ai/a⇤i

t0 i

Figure 4: Optimal trade taxes for the manufacturing case. The left panel assumes no tradecosts, d = 0. The right panel assumes trade costs, d = 1.44.

policy matches the U.S. manufacturing import share—i.e., total value of U.S. manufactur-ing imports divided by total value of U.S. expenditure in manufacturing—as reported inthe OECD STructural ANalysis (STAN) database in 2009, 24.7%.

Results. Figure 4 reports optimal trade taxes as a function of comparative advantage formanufacturing. As before, the left and right panels correspond to the models withoutand with trade costs, respectively, under a normalization with zero import tariffs. Likein the agricultural exercise of Section 6.1, we see that the maximum export tax is close tothe optimal monopoly markup that a domestic firm would have charged on the foreignmarket, s/ (s � 1)� 1 ' 66.7%, suggesting that the U.S. remains limited in its ability tomanipulate the foreign wage.

Table 2 displays welfare gains in the manufacturing sector. In the absence of tradecosts, as shown in the first two columns, gains from trade for the U.S. are 33% larger un-der optimal trade taxes than in the absence of any trade taxes (36.85/27.70� 1 ' 0.33) and86% smaller for the R.O.W. (1� 0.93/6.59 ' 0.86). This again suggests large inefficienciesfrom terms-of-trade manipulation at the world level. Compared to our agricultural ex-ercise, the share of the U.S. gains arising from the use of non-uniform trade taxes is noweven larger: more than two thirds (30.09/27.70 � 1 ' 0.09).

As in Section 6.1, although the gains from trade are dramatically reduced by tradecosts—they go down to 6.18% and 2.02% for the U.S. and the R.O.W, respectively—theimportance of non-uniform trade taxes relative to uniform tariffs remains broadly sta-ble. In the presence of trade costs, gains from trade for the U.S., reported in the thirdcolumn, are 49% larger under optimal trade taxes than in the absence of any trade taxes

29

Gains from Trade

Gains from Trade

No Trade CostsNo Trade Costs Trade CostsTrade Costs

U.S. R.O.W. U.S. R.O.W.

Laissez-Faire 27.70% 6.59% 6.18% 2.02%

Uniform Tariff 30.09% 4.87% 7.31% 1.31%

Optimal Taxes 36.85% 0.93% 9.21% 0.36%

Concluding Remarks

• First stab at how CA affects optimal trade policy

• Simple economics: countries have more room to manipulate prices in their CA sectors


• Larger subsidies are not about desire to expand, but constraint on ability to contract

Concluding Remarks

• More applications of our techniques ≠ market structures (e.g. BEJK, 2003; Melitz, 2003)

• Results suggest design and gains from trade policy depends on micro-level heterogeneity

comparative advantage and optimal trade taxes - … · comparative advantage and optimal trade...

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