1

Import Tariffs and Growth in a Model with Habits
Been-Lon Chen, Shun-Fa Lee and Koji Shimomura

1. Introduction
This paper studies the relationship between tariffs and economic growth in an endogenous
growth framework. While most existing documents find a negative relationship between importtariffs and the rate of economic growth, there are works finding an ambiguous, or even positive,
relationship.
In existing theoretical works , most studies maintained that tariffs slow down economic growth(e.g., Jones and Manuelli, 1990; Easterly and Rebelo , 1993; Osang and Pereira, 1996; andBen-David and Loewy, 1998). However, others held different points of view. In a R&D model,
for example, Grossman and Helpman (1990) argued that the relationship between tariffs andeconomic growth is ambiguous , depending on how the fraction of resources to the R&D sector isaffected. Similar argument has been made in a R&D model by Rivera-Batiz and Romer (1991).1
In empirical works, while most papers documented a negative relationship between tariff rates
and economic growth (e.g., Lee, 1993; Harrison, 1996; and Edwards, 1998), Rodriguez and Rodrik(2001) and Sala-i-Martin (1997) contended that the relationship is not robust. Indeed, Yanikkaya(2003) has found a positive relationship using cross-country data over the period of 1970–1997.
Moreover, O’Rourke (2000) and Irwin (2002) reported a positive relationship between tariffs andeconomic growth in the late 19th and early 20th century, as opposed to the otherwise consensus ofa negative relationship in the Post-War era.

Recently, Clemens and Williamson (2004) haveconfirmed that high tariffs were associated with fast growth before World War II, but with slow
growth thereafter.
We build a dynamic trade model in a two-country (Home and Foreign), two-output (pureconsumption and investment goods) Ricardian framework with capital accumulation as the sourceof economic growth, as opposed to R&D as the source of economic growth in the Grossman andHelpman (1990) and Rivera-Batiz and Romer (1991) model.

In this paper, we use Ak

technologies in production. The choice is made in order to differentiate the role of physical1

Rivera-Batiz and Romer (1991) show that the growth rate is a non-monotone function of the tariff rate: it
first declines as the tariff rate rises from zero, and then rises after a positive critical tariff rate, albeit thegrowth rate in the free trade regime is never reached.

2capital accumulation from the human capital accumulation as a source of economic growth. 2

A

special feature of our model is to incorporate time non-separable preferences; we introduce theendogenous rate of time preference and the average living standard in the foreign as benchmark
consumption.
A growing body of empirical evidence using country or cross-country data has confirmed theimportance of time non-separable preferences.3

In particular, an extensive literature on asset

pricing and real business cycles lends further credit to the level of benchmark consumption being asignificant determinant of consumption behavior.4

Habit-forming consumers dislike large andrapid cuts in consumption. While the endogenous rate of time preference equalizes the marginal
products of capital to the subjective rate of discount across countries in order to avoid sudden
capital flight following a shock, the average living standard results in a large premium to hold risky
assets that might force a rapid cut in consumption.
Despite this evidence supporting the relevance of benchmark consumption levels for current
consumption decisions, no attempts have been made to introduce time non-separable preferences
into endogenous growth literature in an open economy framework. Recently, this approach has
been pursued by Carroll et al. (2000), Fisher and Hof (2000), Alonso-Carrera et al. (2005) and
Chen (2006) in an endogenous growth, closed-economy framework. 5

Our model may beconsidered as an extension of these works to an open-economy framework. In particular, weinvestigate the effects of tariffs on the trade patterns and the sustainable rate of economic growththat have never been studied in these existing works.
Our main findings may be summarized as follows.

First, the Ricardian theorem ofcomparative advantage holds in the two-country world even if the preferences are different acrosscountries. Second, in the case of incomplete specialization, local indeterminacy emerges under
milder conditions, as compared to Drugeon (1998) and Nishimura and Shimomura (2006).

Third,

in the case of complete specialization, the relationship between tariffs and economic growth is2

The introduction of human capital accumulation needs a third sector that is non-tradable, pure investment
goods. This makes the analysis complicated without adding the value.
3

See Van de Stadt et al. (1985) using data in the Netherlands, Osborne (1988) using data in the U.K., andFurther and Klein (1998) using data in the G-7 countries.
4

See Abel (1990), Constantinides (1990), Campbell and Cochrane (1999) on asset pricing and Ljungqvist
and Uhlig (2000) and Boldrin et al. (2001) on real business cycles.
5

In contrast, there is a number of works in a bounded growth framework. Examples include Dupor andLiu (2003), Alonso-Carrera et al. (2004, 2006), Liu and Turnovsky (2005), among others.

3

ambiguous. When the Home country specializes in the investment (resp. consumption) goods, asufficiently higher rate of tariffs on the consumption (resp. investment) goods reverses the tradepattern in the long run and decreases economic growth when the productivity coefficient of theinvestment goods in the Home (resp. Foreign) country is larger than the threshold.

However,

economic growth is increased when the productivity coefficient of the investment goods in the
Home (resp. Foreign) country is smaller and in the Foreign (resp. Home) country is larger than thethreshold. Finally, tariffs increase (resp. decrease) the long-run welfare in the Home country
when it specializes in the investment (resp. consumption) goods.
Intuitively, the effects of a tariff on economic growth work through its effect on the difference
between the interest rate and the rate of time preference.

The import tariff raises the price of theimported goods in the Home country, attracting more resources into this sector. As a result of ahigher demand for capital, the interest rate is increased in the Home country.

However, theinternational price of the imported goods is reduced by augmented output supply, thereby
decreasing the demand for capital and the interest rate in the Foreign.

Higher domestic and lower

foreign interest rates indicate that the ratio of Home to Foreign consumption increases (resp.
decreases) when the Home country is specialized in the investment (resp. consumption) goods,
resulting in a higher (resp. lower) rate of time preference in the Home country. Moreover, thechange in the interest rates in the Home and Foreign countries depends on the productivity level inthe investment goods sector. When the productivity level of the investment goods in the Home
(resp. Foreign) country is higher than the threshold, the increase in the interest rate is smaller than
the increase in the rate of time preference. As a result, economic growth is lower. When the
productivity level of the investment goods in the Home (resp. Foreign) country is smaller than thethreshold and in the Foreign (resp. Home) country is larger than the threshold, the former makes
the interest rate increase more, whereas the latter makes the rate of time preference increase less.
As a result, the rate of economic growth is increasing in tariffs.
Finally, the reason for the effect on long-run welfare is simple. The long-run welfare in the
Home country is increasing in the ratio of consumption between the Home and Foreign country.
When the Home country specializes in the investment (resp. consumption) goods, tariffs on theimported consumption (resp. investment) goods increase (resp. decrease) the ratio of consumption
between the Home and Foreign country in the long run, thereby increasing (resp. decreasing)long-run welfare in the Home country.
The structure of the paper is as follows. In section 2, we set up the basic model.

In Section

4

3, we analyze the balanced growth path and the transit ional dynamics.

In Section 4, we study the

relationship between the import tariff policy and economic growth.

In Section 5, we investigate

the welfare effect of tariffs. Finally, some concluding remarks are made in Section 6.

2. The Basic Model
2.1. Firms
There are two sectors in each country: a pure consumption good and a pure investment good
sector; both use only capital, k i.6

The capital may be thought of as a composite of various types ofphysical and human capital as outlined in Rebelo (1991).

Following the Oniki and Uzawa

tradition, we assume that while the two goods are tradable, capital stock is not internationallymobile.

The production function of good i is denoted asyi = ai ki , i=1, 2.

where i=1 (resp. 2) corresponds to the consumption (resp. investment) good.

(1)
The full

employment conditions arek = k1 + k2 ,(2)where k is the total capital stock.
The first-order conditions for Home’s representative competitive firm are
pa1 = r ,(3a)a2 = r ,(3b)where p is the price of consumption good in terms of the investment good; r is the interest rate , andthus the net rental rate of capital if we assume that there is no depreciation of capital.
There are two countries in the world , Home and Foreign, which may have different production
technologies and preferences. Suppose the Foreign has a similar production structure to those in(1) and (2). Denote the variables with an asterisk as in the Foreign. Then this is a Ricardianmodel, which means that the trade pattern is determined by comparative advantage. Thus, weclassify into three types of the trade pattern.
Case 1: a 1 /a 2 =a 1 */a2 *This is the case of incomplete specialization with each country producing both goods.
Conditions (3) imply that both p and r are constant; i.e. p=a 2 /a 1 and r=a 2 .
Case 2: a 1 /a 2 <a1 */a 2 *6With only one consumption good, the quadratic utility form used in existing studies can be relaxed.

5Complete specialization arises in this case. The Home country only produces and exports theinvestment goods and imports the consumption goods that is produced solely by the Foreign; hence,
k1 =0, k 2 =k, k* 1 =k* and k 2* =0.

Therefore, Conditions (3) indicates that r=a2 and r*=a 1 *p.

Case 3: a 1 /a 2 >a1 */a 2 *In this case, the trade pattern is reversed and thereby k1 =k, k 2 =0, k 1* =0 and k* 2 =k*. Thus,
Conditions (3) indicates that r=a1 p and r*=a2 *.
In sum, combining (2) and (3), we obtaink = k , k = 0; k * = k 2* , k1* = 0; r = a p; r* = a 2* ; a / a < p < a * / a * ;a1 / a2 > a1* / a*2
1 2
1
2
1
2
1

k * = k1* + k2* ;r = a 2 ; r* = a *2 ;p = a 2 / a1 ;if a1 / a2 = a1* / a2* . (4) k = k1 + k 2 ;k = k , k = 0; k * = k * , k * = 0; r = a ; r* = a * p; a / a > p > a * / a * ;a1 / a2 < a1* / a*2
2 1
2
1
2
1
2
1
2
1


2.2 Households
The representative household earns factor income with k(0) units of capital endowed initially.
The household’s budget constraint is
k& = rk − pc ,(5)

where c is consumption.
The felicity of the representative agent is u(c, c* ), where c* is the average consumption level inthe foreign country, a consumption externality.

The felicity allows for the idea that the

representative household cares about his living standard relative to the average living standard in
the Foreign.

We assume that u(c, c* ) is increasing and concave in c. Moreover, we assume thatu(c, c* ) is decreasing in c* . The assumption is similar the one made in Abel (1990) and Dupor andLiu (2003) in which an individual is keeping up with the Joneses.
The representative agent maximizes the discounted sum of the lifetime felicity, with discountfactor X. The discount factor changes in the following way;
X& = − ρ (u ) X .

(6)

Following Uzawa (1968), we call ρ(u) the time preference function and consider
Assumption 1 0 < ρ (0) < a2 and ρ ( u) > 0, ρ ′( u) > 0, ρ ′′(u ) > 0, ρ (u )− u ρ ′(u ) > 0 for all u > 0.
We use the following form for the felicity,u (c , c*) = u ( cc* ).

(7)

While the above form is consistent with the assumption of a positive and decreasing marginal

6

utility in c and a decreasing utility in c* , let us make some remarks. In Abel (1990), Gali (1994),
Chen (2006) and others, the form u[ (c c*)γ ] is used, where 0<?=1. Our form is a special case ofthis class of felicity that takes the extreme case at ?=1.

The reason for choosing ?=1 is due to theendogenous rate of time preference in an open economy with perpetual growth as follows.
In a two-country dynamic Heckscher-Ohlin model by Baxter (1992), the responses to taxes
and other shocks are very dramatic and not continuous.

Chen, et al. (2005) finds that introducing

an endogenous rate of time preference avoids such a problem and assures continuous responses.
In a recent paper with endogenous rates of time preference Palivos et al. (1997) showed that therate of time preference must be constant along a Balanced Growth Path (BGP).

Given that therate of time preference is monotone in the felicity, the only way to guarantee a constant rate of time
preference along a BGP is ?=1. Moreover, our form (7) is consistent with the recent paper on
endogenous rates of time preference by Drugeon (1998) that shows that the homogeneity of degree
zero with respect to an individual consumption and the living standard is a necessary condition for
the existence of a BGP. 7 Thus, although our function form is restrictive, it is necessary in order
to assure a BGP.

Nevertheless, when we introduce the distortionary tariff policy, the consumption

ratio in a BGP could deviate from unity.
Let z=c/c* . We make the following standard assumptions for the felicity in (7).
Assumption 2 u (0) = 0, u ′( z) > 0, u ′′( z) < 0, [u ′( z ) z ]/ u( z ) < 1.
If we denote λ and µ as the co-state variables of k and X, the Hamiltonian of the household
optimization problem isℑ = u ( cc* ) X + λ{rk − pc} + µ{ − ρ (u )X },The necessary conditions for optimality areu ′( cc* )[1 − µρ ′(u (cc* ))] =7

λ pc *
,
X

(8a)

λ&
= − r,λ(8b)µ&= ρ (u (cc* )) − 1µ u ( cc* ),µ(8c)In a recent paper Drugeon (1998) introduces endogenous discounting that is increasing in consumption
and decreasing in the living standard in a society, measured by the average consumption in the society. Ourformulation is consistent with Drugeon (1998) in that the discounting is decreasing in the average foreign
living standards.

7with the two transversality conditions lim t→ ∞ λ k = 0, and lim t→∞ µ X = 0.
While condition (8a) equates the marginal utility of consumption to the marginal cost offoregone savings, conditions (8b-c) are Euler equations for capital and the discount, respectively.

3. Two-country World Market Equilibrium
The world commodity market-clearing condition for the consumption goods isc + c * = a1k1 + a1*k1*.

(9)

Once (9) is satisfied, the world market for the investment goods is automatically cleared.

3.1. Transformation of the Economic System
In order to analyze the equilibrium, it is necessary to transform the equilibrium conditions
with perpetual growth into a system with stationary variables. Denote m=pc/k , m* =pc* /k * , v1 =k 1 /k,
v2 =k2 /k, v1 *=k1 */k* and v2 *=k 1 */k*. In what follows we briefly explain the transformation.
First, (8c) in terms of the transformed variables and its counterpart for the Foreign areµ&= ρ (u( z)) − µ1 u ( z ),
µ

(10a)

µ& *
= ρ * ( u( 1z )) − µ1 * u* ( 1z ).
*
µ

(10b)

Next, differentiating (8a) and its foreign counterpart, with (8b) (10a) and (10b), yields*µρ ( u ( z )) −u ( z )]c&
= χ 1(z) {r − ρ (u( z)) − [1 − χ ( z )] cc&* − &pp − ρ ′ (u ( z1))[− µρ},c′ (u ( z ))(11a)

ρ ′ ( u ( 1z ))[ µ ρ (u ( z1 ))− u ( z1 )]
p&
*
*
* 1
*
c&*
1
c&
{
r
−
ρ
(
u
(
))
−
[1
−
χ
(
z
)]
−
−
},
* =
*
z
c
p
c
χ (z )
1 −µ * ρ *′ (u * ( 1z ))

(11b)

*

′

*

*

*

*

*

′′

where χ ( z) ≡ σ ( z ) + 1−µρµρ(′(uu( (z ))z)) ψ ( z ) > 0, in which σ ( z) ≡ −uu′ ((zz ))z > 0 denotes the reciprocal of theintertemporal elasticity of substitution in the Home country and′′′ψ ( z) ≡ ρ (uρ ′((zu))( uz))( z ) z > 0, theelasticity of the marginal time preference rate with respect to z in the Home country.
µ * ρ *′ ( 1 )−u *′′ ( 1 )ρ *′′ ( 1z ) u *′ ( 1z )z

ρ *′ ( 1z ) zχ * ( z) ≡ σ * ( z ) + 1− µ *ρ *′ (z 1 ) ψ * ( z ) > 0, σ * ( z) ≡ u*′ ( 1 )zz and ψ * ( z) ≡z

counterparts in the Foreign country.
Then, if we substitutep&pin (11a) into (11b), we obtain

Similarly,

are the corresponding

8

{

z&
1
=
r − ρ (u( z)) − r* + ρ * (u * ( 1z ))*z χ ( z ) + χ ( z )− 1
*
*
* * *
*
ρ ′(u( z))[ µρ (u( z)) − u ( z )] ρ ′ (u ( 1z ))[µ ρ (u ( 1z )) − u ( 1z )] −+.
1 − µρ ′(u( z ))1 − µ * ρ *′ (u * ( 1z ))


(12)

Moreover, (5) is rewritten ask&= r − m.
k

(13)

Using (11a)-(13), we obtain&
ρ *′ ( u* ( 1z ))[ µ * ρ *(u * ( z1 )) − u * ( z1 )]mz&= χ * ( z ) + r * − ρ * (u * ( 1z )) −− r + m,mz
1 − µ * ρ *′ (u * ( 1 ))

(14a)

m
&*
z&
ρ ′(u ( z ))[ µρ (u( z)) − u ( z )] *= − χ ( z) + r − ρ (u( z)) −− r + m* .
*′z
1 − µρ (u ( z ))
m

(14b)

z

Further, dividing (9) by c*, the world market-clearing condition may be rewritten asz +1= p(a1z v1a*v *+ m1 *1
m

).

(15)

Finally, the full employment conditions and the first-order conditions for the firm summarizedin (4) can now be rewritten as follows.
 v = v * = 1; v 2 = v1* = 0; r = a p; r* = a*2 ; a / a < p < a * / a *;a1 / a2 > a1* / a2*
1
2
1
2
1
2
1

*
*
*
*
p = a2 / a1 ;if a1 / a2 = a1* / a2* .
 v1 + v2 = 1; v1 + v2 = 1; r = a2 ; r = a2 ;

*
*
*
*
*
*
a1 / a2 < a1* / a2*
 v2 = v1 = 1; v1 = v 2 = 0; r = a2 ; r = a1 p a2 / a1 > p > a 2 / a1 ;(16)

Equations (10a), (10b), (12) and (14a)-(16) are the transformed equilibrium system. The
system determines the equilibrium paths of the twelve control variables, z, m, m* , µ, µ* , p, v1 , v2 ,v1 *, v2 *, r and r*.

3.2. Balanced Growth Path
We now analyze the equilibrium in a steady state. A steady state is a perfect foresightequilibrium with a balanced growth path (BGP) under which z, m, m* , µ, µ* , p, v1 , v2 , v1 *, v2 *, r
and r* are constant, and thus

z& / z = m& / m = m& * / m* = p& / p = µ& / µ = µ& * / µ * = 0.

Denotez%, m% , m% * , µ% , µ% * , p% ,v%1 ,v%2 ,v%*1 , v%*2, r% and r%* as the values in a BGP. Then, they are determined by thefollowing relationships.
ρ[u ( z% )]µ% = u ( z%),

(17a)

9

ρ *[u * ( 1%z )]µ% * = u * ( z1% ),(17b)r% − ρ [u ( z% )] = r% * − ρ *[u * ( 1z% )],

(17c)

%,
r%* − ρ *[ u* ( 1z% )] = r% − m(17d)r% − ρ [u( z%)] = r%* − m% * ,(17e)a z%v%
a v%
z% + 1 = p% 1m% 1 + m1% *1 ,

(

(17f)

1 = v%1 + v%2 , 1 = v%1* + v%2* ,

(17g)

* *

)

 v%2 = v%1* = 0,

if a1 / a2 > a1* / a*2 ,

*
a2
a2
*
*
 p% = a1 = a1 * , if a1 / a2 = a1 / a2 ,*** v%1 = v%2 = 0, if a1 / a 2 < a1 / a2 ,
 a1 p% ≥ a2 , if a1 / a2 ≥ a1* / a2* , *  a*2 ≥ a1* p% , if a1 / a2 ≥ a1* / a2* ,r% = 
r% =  **** a2 ≥ a1 p% , if a1 / a2 ≤ a1* / a2* , a1 p% ≥ a2 , if a1 / a 2 ≤ a1 / a2 .

(17h)

(17i)

Three cases are in order to analyze the existence of BGP.
Case 1: a 1 /a 2 =a 1 */a2 * (Incomplete specialization.)First, (17h) and (17i) indicate that p% = a2 / a1 , %r =a2 and %r* = a2* and are all constant.
Substituting them into (17c) yields
a 2 − ρ [u ( z% )] = a2* − ρ * [u*( 1%z )](18a)that determines a unique z% under Assumptions 1 and 2.
Next, if we substitute z% into (17a), (17b), (17d) and (17e), µ% , µ% * , m% , and m% * are determineduniquely, respectively.

% and m% * , the three relationships in (17f)
Finally, if we substitute in z%, mv%1, v%2 , v%1* and v%2*.

and (17g) need to determine the four endogenous variables:
BGPs thus emerges.

Multiplicity ofGiven a set of values for v%1 and v%1* , then v%2 and v%2* can be uniquelydetermined by the two relationships in (17g).
Using (17f), the set of values for v%1 and v%1* is determined by

{

(

* *

)}

( v%1 ,v%1* ) = ( v%1 ,v%1* ) | z% + 1 = p% a1mz%%v%1 + am1%v%*1 .

It is obvious that there is a continuum of BGPs, indexed by v%1. For each v%1 , there exists a

10

unique v%1* that satisfies the equilibrium condition in the above expression.
value of v%1 in (0,1) is free, there is a continuum of BGPs.

As the choice of the

The results indicate that any

combination of incomplete specialization that satisfies the market-clearing conditions in the twocountries is a long-run equilibrium. As both v%1 > v%1* and v%1 < v%1* could emerge, the long term
trade patterns are thus indeterminate for these two countries.
Case 2: a 1 /a 2 <a 1 */a 2 *. (Complete specialization.)In this case, the Home country completely specializes in the production of and exports theinvestment goods, while the Foreign country completely specializes in and exports theconsumption goods.
Substituting (17h) and (17g) into (17f), we obtain a1 p% /( z% + 1) = m% * .

This relationship,

together (17c), (17e) and (17i), yields
a 2 − ρ ( u( z% )) = ρ * (u * ( 1z% )) z%.

(18b)

The left hand side of (22) is decreasing in z% and the right hand side of (18b) is increasing in z%
under Assumptions 1 and 2. 8Thus, (18b) determines a unique z%.

r%, r%*, µ% , µ% * , m% and m% * are uniquely determined by the other equations.
v%1 = v%2* =0 and v%2 = v%1* = 1.

Then, variables
In particular,

Using (17i), the equilibrium relative price of the consumption goods inthe long run is therefore uniquely determined as p% = r%* / a1* .
Case 3: a 1 /a 2 >a 1 */a 2 *. (Complete specialization.)In this case, the Home country completely specializes in the production of and exports theconsumption goods, while the Foreign completely specializes in and exports the investment goods.
First, we use (17c), (17d), (17f), (17h) and (17i) to obtaina *2 − ρ *[u * ( 1z% )] =

ρ [u (z% )].
z%Assumptions 1 and 2 assure the existence of a unique z%.

(18c)Then, other variables are determined in

the fashion similar to the way used in Case 2. The equilibrium relative price of the consumption
goods is p% = r% / a1 .
We should mention that in a BGP, the rates of economic growth for consumption and income8∂ ( ρ *( u * ( 1 )) z% ) = ρ * − u * ρ *′ ( u*′ ) > 0, and ∂ ( ρ ( u( z)) ) = −[ ρ −u ρ′ ( u′z / u )] < 0,
∂z
z%
∂z
z
u*z%
z2

under Assumption 1 and 2.

11

are equal.

The long-run rate of economic growth is determined as follows.

In a BGP,

p&
= 0,
p

and moreover, (17a) and (17b) indicates µ% = u / ρ and µ% * = u* / ρ* .

Substituting these relationships into (11a) and (11b), the economic growth rates in the twocountries are respectively

Then, if we substitute

c&*
c*

*
c&
= χ 1(z ) {r − ρ (u ( z )) − [1 − χ ( z)] cc&* },
c

(19a)

c&*
= χ *1( z ) {r − ρ (u ( z )) − [1 − χ ( z)] cc& }.
c*

(19b)

in (19b) into (19a), we obtainc&= [ χ +χχ**−1] {r − ρ − 1χ−*χ (r * − ρ *)}.
cUsing (17c), we then obtainc&= r − ρ [u( z )].
c

(20a)

c&*
= r * − ρ *[u * ( z1 )].
c*

(20b)

In a similar fashion, we obtainTo summarize the results analyzed above, we obtain
Proposition 1

Under AK technologies and Assumptions 1 and 2, the Ricardo theorem ofcomparative advantage holds in the two-country world even if the preferences are different acrosscountries.
Remark 1

The model is a Ricardo two-country endogenous growth model.

consumption is determined by the production technology.

The price ofThus, the trade pattern may be changedby the distortion of the firm’s behavior but not of the household’s behavior. For examples, output
taxes or import tariffs may affect the trade pattern.
Remark 2

When the two countries have the same technologies, the factor price equalization

theorem holds in the case of incomplete specialization, in which r=a 2 =a 2 *=r*, but fails in the caseof complete specia lization with exception only when the following conditions are met.
 r ≡ [a *2 − ρ * (u* ( 1 ))](1 + z% ) = a2* ≡ r* , if a1 / a2 > a1* / a2* .
z%

*
1+ %z
if a1 / a 2 < a1* / a*2 .
 r ≡ a2 = [a2 − ρ (u( z%))]( %z ) ≡ r ,3.3. Local Dynamics

12

In this subsection, we analyze the local dynamics. We only study the situation when bothcountries are incomplete specialization; i.e., under Case 1 where a1 /a2 =a 1 */a 2 *.
It is obvious that the dynamic system is recursive: While (10a), (10b), (12), (14a) and (14b)simultaneously govern the dynamics of z, m, m* , µ and µ* , the dynamics of other variables areeasily determined by substituting these variables into other equations.

Note that although there isa continuum of equilibrium values for v%1 , v%2 , v%1* and v%*2 , the equilibrium values for z, m, m* , µ andµ* , together p% , r% and r% * , are unique along a BGP. We consider
Condition D: σ +µ * ρ *′µρ ′
ψ > 1 and σ * +ψ * > 1.
**′
1 − µρ
1− µ ρ ′Condition D requires a sufficiently small intertemporal elasticity of substitution; namely, asufficiently large value of s.9
If we apply linear Taylor expansion of (10a), (10b), (12), (14a) and (14b) near the unique( µ% , µ% * , z% ,m% ,m% * ), we obtain0 µ&   ρ µ& *   0 ρ *   z&  =  b31 b32&   b41 b42
m
* 
m
 &   b51 b52

where

b13
b23
0
0
0

0 0   µ − µ% 
0 0   µ * − µ% * 
0 0   z − z%  ,% % 0  m − mm*  *% * 
0 m%   m − m(21)b13 = −u ′(1 − µρ ′) < 0,b23 = uz*2′ (1 − µ * ρ *′ ) > 0,
b31 =

b32 =
b41 =

b42 =

9

−ρ ′ρ z< 0,−1)(1 − µρ ′)(χ + χ*ρ *′ρ * z
( χ + χ * −1)(1 − µ * ρ *′ )> 0,

− χ *m ρ ′ρ< 0,
( χ + χ * −1)(1 − µρ ′ )− m ( χ − 1)ρ *′ρ *( χ + χ * −1)(1 − µ * ρ *′ )< 0,

It is in general documented that s is larger than 1, so Condition D is easily met. Ogaki and Reinhart(1998) estimated the value using the US data and obtained σ∈[2.22, 3.125]. Yogo (2004) estimated datausing cross-country data and uncovered σ>2 across eleven developed countries which was smaller than σ>5
he estimated for the U.S. economy.

13

b51 =

b52 =

− ρ ′ρ m * ( χ * − 1)< 0,
( χ + χ * −1)(1 − µρ ′)− m* χρ *′ρ *( χ + χ * −1)(1 − µ * ρ *′ )< 0.

Under Condition D, we obtain b 13 <0, b 23 >0, b 31 <0, b 32 >0, b 41 <0, b 42 <0, b 51 <0, and b 52 <0.
Notice that the system in (21) only includes variables that may jump instantaneously and does
not involve any state variables whose values are predetermined. As a result, there exists a uniqueequilibrium path toward the BGP if the number of eigenvalues with negative real parts for theJacobean matrix on the right hand side of (21) is zero. On the other hand, if the number of
eigenvalues with negative real parts is larger than or equal to one, then there exists a continuum ofequilibrium paths toward the BGP.
Denote by J the Jacobean matrix in (21), by θ the corresponding eigenvalues and by I an
identity matrix of order 5. The eigenvalues are then determined by% − θ )(m% * − θ )Γ(θ ) = 0,
Ω (θ ) ≡ J −θ I = ( mwhere Γ(θ ) = {−θ 3 + (ρ + ρ * )θ 2 − ( ρρ * − b13b31 − b23b32 )θ − (b13b31 ρ* + b23b32 ρ )}.
It is clear to see that two of the five eigenvalues are θ1 = m% > 0 and θ2 = m% * > 0.
remaining three roots, denoted by ?3 , ?4 and ?5 , are determined by G(?).

The

Using the result thatΓ(0) ≡ θ3θ4θ5 = −( b13b31ρ * + b23b32 ρ ) < 0,there are either one or three roots with negative real parts.
equilibrium path toward the BGP is local indeterminacy.

As a consequence, the dynamic

To summarize,

Proposition 2. Under Condition D, suppose that the two countries in the world have technologies
with identical comparative advantage, i.e. a 1 /a 2 =a1 */a 2 *. Then, the BGP is locally indeterminate.
Our result may be compared with that of Nishimura and Shimomura (2006), which finds the
emergence of local indeterminacy in the neighborhood of a steady state in a similar two-country
trade model with bounded growth.10

The local indeterminacy in our model with unbounded

growth is established with weaker requirements. Our model shares with that of Nishimura and10

In Nishimura and Shimomura (2006), the emergence of local indeterminacy requires that (i) the
technology be of Leontief-type and the explicit utility form be quadratic, (ii) the consumption good be morelabor intensive than the investment good and (iii) parameter values for technology and preference are
restricted.

14

Shimomura (2006) in the use of an AK technology that is similar to a Leontief technology.
Different from these authors, we use an implicit functional form for the felicity by assuming only
one kind of consumption goods. As a result, the emergence of local indeterminacy in our model
requires neither a particular factor intensity ranking nor specific restrictions on parameters, except
for a sufficiently small intertemporal elasticity of substation. However, there is an external effectin the felicity in our model emerging from the consumption standard in the Foreign that is absent inNishimura and Shimomura (2006).

Because of the external effect, restrictive conditions necessary

to obtain indeterminacy in Nishimura and Shimomura (2006) are not required in our model.
The external effect is reminiscent of the consumption externalities in a closed-economy,
unbounded growth model by Drugeon (1998). By assuming an agent’s rate of time preference
that is decreasing in the consumption standard measured by average consumption, Drugeon (1998)
finds the emergence of local indeterminacy.

Like Drugeon (1998) the rate of time preference isdecreasing in the consumption externalities in our model. Yet, different from Drugeon (1998), thefelicity is decreasing in the consumption externalities in our model. As a result, the consumption
externalities may be likely to offset the possibilities of the emergence of local indeterminacy via thenegative effect on the felicity.

Because of the open economies in our model, the emergence of

local indeterminacy does not require the former effect to dominate the latter effect.

4. Import Tariff and Growth
In this section, we examine the long-run relationship between import tariffs and growth.

To

simplify the analysis, we assume two countries identical in every aspect, except for thetechnologies. As a result, ρ * ( u*( Z1 )) = ρ (u ( Z1 )).

As the case of incomplete specialization (Case 1)

involves a continuum of BGPs, our comparative-static analysis is conducted under the case ofcomplete specialization (Cases 2 and 3) that has a unique BGP.

4.1 Home country specializes in the investment goods
This is under Case 2 in Section 3.2, in which a 1 /a 2 <a 1 */a2 *, and thereby, the Home country
exports the investment goods and imports the consumption goods.
Suppose that the government in the Home country imposes an ad-valorem import tariff on the
consumption goods with the amount of tariff revenue transferring to the households in a lump-sum

15fashion. 11

The government budget constraint is
T = spT ( c − y1 ),(22a)where s is the tariff rate of the imports, p T is the international price of the consumption goods under
the tariff policy, and T is the transfer payment. In what follows, a variable with subscript T is used
to denote the variable under the tariff regime, while a variable with subscript F is used to denotethe variable under the free trade regime.
As the Home country completely specializes in investment goods, y1 =0 and (20a) becomes
T=spTc. Now, p T(1+s) is the Home’s domestic prices of the pure consumption good.

Thus, theflow budget constraint of the household is
k& = rk − pT (1 + s) c + T .

Note that m= p(1+s)c/k when (1+s)a 1 /a 2 >a 1 */a2 *.

(22b)

In a BGP, equilibrium system (17) remains

the same, except for (17f), (17h) and (17i) that now become()

(1+s )a1zv% %1
a1*v%1*
%
z
%
+
1
=
p
+
, if (1 + s)a / a ≥ a* / a * ,
T

m%
m
%*
1
2
1
2

* *
*
*
 z% + 1 = p% a1z% v%1 + a1 v%1 , if (1 + s)a1 / a2 ≤ a1 / a2 ,
T
m
%
m
%
*


(

)

(17f)'

 v%2 = v%1* = 0,
if (1 + s) a1 / a2 > a1* / a2* ,

*
a2
a2
*
*
(17h)'
 p% T = (1+ s) a1 = a1* , if (1 + s) a1 / a2 = a1 / a2 ,*if (1 + s) a1 / a 2 < a1* / a2* , v%1 = v%2 = 0,**
a1 p% T (1 + s ) ≥ a 2 , if (1 + s )a1 / a2 ≥ a1 / a2 , *  a*2 ≥ a1* p% T , if (1 + s)a1 / a 2 ≥ a1* / a2* ,r% = 
(17i)'
r% =  ****a 2 ≥ a1 p%T (1 + s), if (1 + s )a1 / a2 ≤ a1* / a*2 , a1 p% T ≥ a2 , if (1+ s) a1 / a 2 ≤ a1 / a2 .

To analyze the impacts of import tariffs on economic growth in a BGP, we compare theeconomic growth rate under tariff with the rate under free trade.

Denote sˆ ≡ a1a*/1 /aa22* − 1.

Apparently, sˆ > 0 in the case under study.
First, if s < sˆ, then (1 + s) a1 / a2 < a1* / a2* , and thereby, the trade pattern is not changed.
The rate of economic growth depends upon the value of z% along the BGP. As z% is not affected,
the rate of economic growth is study.
Second, if s > sˆ, then (1 + s )a1 / a 2 > a1* / a2* .
11As a result, the trade pattern is reversed.

The results under a specific import tariff are qualitatively similarly to those of an ad-valorem import tariff,
but the analysis under an ad-valorem import tariff is simpler. We thus focus on the analysis of the effects ofan ad-valorem import tariff.

16

The economy changes from the importer to the exporter of the consumption and imports theinvestment goods.

The tariff is on the imports of consumption goods, but the Home country

imports the investment goods after the tariff policy is enforced. As a consequence, the Homecountry does not have any tariff revenues, and thus, the transfer payment is zero in the long-run
equilibrium.
To investigate the effect of the tariff policy on the interest rate, we note from (17i) that under
the free trade regime r%F = a 2 and r%F * = a1* p% F .

Because of the tariff policy, there is an excess of

supply of the consumption goods in the world market at p% F , making the world-market clearing
price lower, p% T < p% F . Thus, after the policy is enforced, (17i)' indicates that r%T = a1 p% T (1 + s ) > a 2
and a1* p% F > r%T * = a *2 > a1* p% T .

Therefore, while the interest rate in the Home country is raised

under the tariff regime, i.e., r%T > r%F , the interest rate in the Foreign is lowered under the tariff
regime, i.e., r%T * < r%F *. As a result, (17c) suggests z%T > z%F .

Moreover, (18b) indicates that≥z%F ≥< 1, if a2 < 2 ρ ( u (1)),

(23a)

*≤
z%T ≥
< 1, if a2 > 2 ρ( u(1)).

(23b)

while (18c) indicates that

We are ready to examine the effect of the tariff policy on economic growth. Under free trade,from (17i) and (20a), the economic growth rate is equal to g F ( z) = a 2 − ρ (u( z )), which using(18b) is rewritten asg F ( z%F ) ≡ ρ ( u( %z1F )) z%F .

(24a)

Under the tariff regime, on the other hand, the economic growth rate in the Home country is
g T = a1 p% T (1 + s) − ρ (u( z )), which using (17c), (17d) and (17f)' is rewritten asgT ( z%T ) ≡ ρ (u ( z%T ))/ z%T .

(24b)

Under Assumptions 1 and 2, g F is increasing in z%F , while g T is decreasing in z%T .
Moreover, Loci g F and g T intersect at z%F = z%T = 1. (See Figure 1.)[Insert Figure 1 here]1.

a 2 ≥ 2 ρ ( u (1)).

In this case, (23a) indicates that the ratio of Home to Foreign consumption is z%F ≥ 1 underfree trade (see point F1 in Figure 1). Recall that the import tariff policy leads to a higher ratio of17Home to Foreign consumption; i.e., z%T > z%F . (Say at Point T1 in Figure 1.)gT ( z%T ) < g F ( z%F ).

2.

As a result,

The rate of economic growth is lower under the tariff regime.

a 2 < 2 ρ( u(1)).

In this case, z%F < 1 under free trade according to (23a). Under the tariff regime, z%T > z%F .
It is possible that economic growth is higher under the tariff regime, depending upon the relative
magnitude between a2 * and 2?(u(1)).
2.1. If a *2 ≥ 2 ρ ( u (1)), (23b) indicates that z%T ≤ 1.

As z%T > z%F , it indicates z%F < z%T ≤ 1.

(See F2 for free trade and T2 for the tariff regime in Figure 1.)

As a result, growth rate is increased

under the tariff regime.
2.2. If a *2 < 2 ρ( u(1)), (23b) suggests z%T > 1, say at T1 in Figure 1.
be z%F < 1 < z%T .

As a result, it must

In this situation, the economic growth is ambiguous after the tariff. Therefore,

the growth effect is ambiguous in this case.
We summarize the results in these different cases in Table 1.
[Insert Table 1 here]
The effects on economic growth can be understood by the effect of the tariff policy on theinterest rate relative to the rate of time preference in the Home country a la (17c).

The importtariff makes the price of consumption goods higher in the Home country, thereby attracting moreresources into this sector. As a result of higher demand for capital, the interest rate is increased in
the Home country.

However, the international price of the consumption goods is reduced byaugmented output supply, thereby decreasing the demand for capital and the interest rate in theForeign.

As the result of higher domestic interest rates and lower foreign interest rate s, (17c)indicates that the ratio of Home to Foreign consumption is increased and the rate of time preferencein the Home country is increasing, ρ′(u( z%T ))u′( %zT ) > 0. So, the economic growth rate, r%T − ρ (u( z )),is ambiguous.
In particular, the change in the interest rates is∆r%T * = a*2 − a1* p% F < 0.

∆r%T = a1 p% T (1 + s ) − a2 > 0andIf a 2 is sufficiently large so that a 2 ≥ 2 ρ ( u (1)), then the increase in∆r%T > 0 is smaller than the increase in ρ′(u( z%T ))u′( %zT ) > 0; hence, the economic growth rater% − ρ (u( z )) is reduced. Alternately, if a 2 is small, such that a 2 < 2 ρ( u(1)), then the increase in∆r%T > 0 is large.

Moreover, from (17c), the increment of z depends on the magnitude of ∆r%T * .

If a2 * is sufficiently large, such that a *2 ≥ 2 ρ ( u (1)), the foreign interest rate decreases less and so18

does the increment of z. This causes the domestic time preference rate to increase less. Thus,
r%T − ρ (u( z )) increases in equilibrium, making economic growth higher.

However, if a 2 * issufficiently small, such that a *2 < 2 ρ( u(1)), the foreign interest rate decreases more and so does
the increment of z. This higher domestic time preference rate leads to an ambiguous sign of
r%T − ρ (u( z )).

To summarize the results,
Proposition 3. Suppose that the felicity and the rate of time preference be identical initiallya2 *across countries. Let sˆ ≡ a1a*/− 1 > 0 , so the Home country imports the consumption goods.
1 / a2

Then, the effect of imposing the tariff rate s on the imports of the consumption goods is as follows.
(i)

If s < sˆ, nothing is affected;(ii)

If s > sˆ, the trade pattern is reversed in the long run, and the Home country has ahigher ratio of Home to Foreign consumption anda. lower economic growth when a 2 ≥ 2 ρ ( u (1));
b. higher economic growth when a 2 < 2 ρ( u (1)) ≤ a *2 ;c. ambiguous economic growth when a 2 < 2 ρ( u(1)) and a *2 < 2 ρ( u (1)).

4.2 Home country specializes in the consumption goods
This is Case 3 in Section 3.2, in which a 1 /a 2 >a1 */a 2 *, and thereby, the Home country exports
the consumption goods and imports the investment goods.
Suppose now that the government in the Home country imposes an import tariff on theinvestment goods.

Then, the government budget constraint is
T = τ ( k& − y 2 ),(25a)where t is the rate of tariff.
As the Home country completely specializes in the production of consumption goods, y2 =0
and (25a) becomes T = τ k&.
Home country.

Now (1+t ) is the domestic prices of the pure investment goods in theThus, the flow budget constraint of the household in (5) now turns into
k& = 1+1τ ( rk − pT c + T ).

Similarly, let m=pC/[(1+τ)k] when a 1 /[a 2 (1+τ)]<a1 */a 2 *.

(25b)
As a result, the equilibrium

conditions (17) in a BGP remain the same except for (17c), (17e), (17g), (17h) and (17i) that now

19

become, respectively, 12r%− ρ (u ( z% )) = r%* − ρ * (u* ( 1%z )),1+τ(17c)"r%− ρ (u( z%)) = r% * − m% * ,1+τ(17e)"()


a1z%v%1
a1*v%1***
 z% + 1 = p% T m% + m% * , if a1 /[a 2 (1 + τ )] ≥ a1 / a2 ,

* *
*
*
 z% + 1 = p%T a1z%v%1 + a1v%1 , if a1 /[a 2 (1 + τ )] ≤ a1 / a2 ,
%
%
(1
+
τ
)
m
m
*


(

)

(17f)"

 v%2 = v%1* = 0,
if a1 /[a 2 (1 + τ )] > a1* / a *2 ,
(1+τ )a 2
a2***(17h)"
 p% T = a1 = a1 * , if a1 /[a 2 (1 + τ )] = a1 / a 2 ,***
if a1 /[a 2 (1 + τ )] < a1 / a 2 , v%1 = v%2 = 0,
 a1 p%T ≥ (1 + τ )a 2 , if a1 /[ a2 (1 + τ )] ≥ a1* / a2* , *  a*2 ≥ a1* p% T , if a1 /[a2 (1 + τ )] ≥ a1* / a2* ,r% = 
(17i)"
r% =  ****(1 + τ )a 2 ≥ a1 p% T , if a1 /[ a2 (1 + τ )] ≤ a1* / a2* , a1 p% T ≥ a2 , if a1 /[a 2 (1 + τ )] ≤ a1 / a 2 .

The effects of the tariff policy depends on the rate of the tariff relative to the thresholdτˆ ≡ aa1 1*//aa22 * − 1 > 0. As in the former subsection, if it is smaller than the threshold, the comparativeadvantage is not changed and as a result, the trade patterns and economic growth both are notaffected. Alternatively, if the rate of tariff is larger than the threshold, the comparative advantage
is distorted sufficiently, so the comparative advantage is changed.

Then, the Home country

produces the investment goods and imports the consumption goods in the long run.

Inequilibrium, there is no tariff revenue. Moreover, the domestic interest rate is reduced but theforeign interest rate is raised.

As a result, z decreases.

As in the former subsection, the effect on

economic growth depends upon the magnitude of the productivity coefficients relative to the time
preferences rate in both countries evaluated at an identical level of consumption.

The results are

summarized in Figure 2 and Ta ble 2.
[Insert Figure 2 and Table 2 here]
To summarize the results in this subsection,
Proposition 4. Suppose that the felicity and the rate of time preference be identical initially12

λ p c*The necessary conditions in (8a) and (8b) now becomes u ′( cc* )[1 − µρ ′( u (cc* ))] = X (1T+τ ) andλ&= − 1+rτ .
λ20

across countries.

Let τˆ ≡ aa1 1*//aa22 * − 1 > 0 , so the Home country imports the investment goods.

Then, the effects of imposing the tariff rate t on the imports of the investment goods are as follows.
(i)

If τ < τˆ, nothing is affected;(i)

If τ > τˆ, the trade pattern is reversed in the long run, and the Home country has asmaller ratio of Home to Foreign consumption anda.

lower economic growth when a *2 ≥ 2 ρ ( u (1));

b. higher economic growth when a *2 < 2 ρ( u (1)) ≤ a 2 ;c. ambiguous economic growth when a 2 < 2 ρ( u(1)) and a *2 < 2 ρ( u (1)).

5.

Import Tariff and Welfare
Finally, we turn to examining the relationship between tariffs and welfare in the long run. In

a BGP, the representative agent’s lifetime utility in the Home country is∞

U = ∫0 u (z%)e− ρ (u ( %z)) t dt =u (z% ).
ρ (u( z%))This indicates that the long-run welfare is increasing in z%. 13

Changes in the welfare dependon the goods the Home country imports.
Under Case 2, in which the Home country specializes in the investment goods, theconsumption goods are imported. Tariffs s on the imports of the consumption goods lead to ahigher z% in the long run if s > sˆ.

As a result, the tariffs increase the long-run welfare in the

Home country if s > sˆ.
In contrast, under Case 3, in which the Home country specializes in the consumption goods,
the investment goods are imported.

Tariffs on the imports of the investment goods result in alower z% in the long run if τ > τˆ, It follows that the tariffs decrease the long-run welfare in the
Home country if τ > τˆ,To summarize the relationship between tariffs and welfare in the long run, we obtain
Proposition 5. Let a 1 /a 2 <(resp. >)a 1*/a 2*. Then, a tariff on the imports of the consumption(resp. investment) goods raises (resp. reduces) the long-run welfare in the Home country if the

13

′
′
dU
= u [ ρ −2ρ u] > 0 based on Assumption 1.
dz%
ρ

21

tariff reverses the trade pattern.

6. Conclusion
While most existing documents find a negative relationship between the import tariffs andthe rate of economic growth, there are works finding an ambiguous, or even positive, relationship.
This paper builds a two-country, two-output trade model and studies the relationship between
tariffs and economic growth in an endogenous growth framework. The Ak technologies are usedin order to differentiate the role of physical capital accumulation from the human capital
accumulation as a source of economic growth. The choice leads to Ricardian technologies; yet,the Ricardian theorem of comparative advantage holds in the two-country world even if thepreferences are different across countries.
A special feature of our model is to incorporate time non-separable preferences. A growing
body of empirical evidence using country or cross-country data has confirmed the importance oftime non-separable preferences. We introduce the endogenous rate of time preference.

Anextensive literature on asset pricing and real business cycles lends further credit to the level ofbenchmark consumption being a significant determinant of consumption behavior.

We

incorporate the average living standard in the foreign as benchmark consumption. We find thatin the case of incomplete specialization, local indeterminacy arises in the neighborhood of a BGP.
We examine the relationship between tariffs and economic growth in the case of complete
specialization.

We find an ambiguous relationship.

In particular, when the Home country

specializes in the investment (resp. consumption) goods, a sufficient higher rate of tariffs on the
consumption (resp. investment) goods reverses the trade pattern in the long run and decreases
economic growth when the productivity coefficient of the investment goods in the Home (resp.
Foreign) country is larger than the threshold. However, economic growth is increased when the
productivity coefficient of the investment goods in the Home (resp. Foreign) country is smaller andin the Foreign (resp. Home) country is larger than the threshold. Finally, tariffs increase (resp.
decrease) the long-run welfare in the Home country when it specializes in the investment (resp.
consumption) goods.

22

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Table 1: Import Tariff and Growth Rate ( a1 / a2 < a1* / a2* ) ?
Import Tariffs

Conditions

Variation of Growth Ratea *as < sˆ ≡ a1 a 2* − 1
1 2−

unchangeda2 ≥ 2ρ[u (1)]lower

a2 < 2ρ[u (1)] ≤ a2*
a2 and a2* < 2 ρ[ u(1)]highera *as > sˆ ≡ a1 a 2* − 1
1 2

ambiguous?The preferences and time preference rates are assumed to be the same between two countries.

Table 2: Import Tariff and Growth Rate ( a1 / a2 > a1* / a2* ) ?
Import Tariffs

Conditions

Variation of Growth Rateaa *
τ < τˆ ≡ a1 *2a − 1
1 2

τ > τˆ ≡ a1*2a − 1
aa *
1

2

−

unchanged

a*2 ≥ 2ρ[u (1)]lower

a*2 < 2ρ[u (1)] ≤ a2
a2 and a2* < 2 ρ[ u(1)]higherambiguous?The preferences and time preference rates are assumed to be the same between two countries.

Figure 1: Growth Rates ( a1 / a2 < a1* / a2* )Growth Rate
gF

T2
F1
F2
0

z%F2

T1

z%T2

z%1F = 1

gT

z%1T

z%

Figure 2: Growth Rates ( a1 / a2 > a1* / a2* )Growth Rate
gT
T2
F1
T1
0

z%1T

F2

z%1F = 1 z%T2 z%F2

gFz%