NorthSouth Trade and Tariff Policya
 Author: Kim Suyi
 Publish: Journal of Korea Trade Volume 16, Issue4, p73~103, Nov 2012

ABSTRACT
We analyze the steady state effect of tariffs and subsidies in a dynamic general equilibrium model of NorthSouth Trade with scale invariant growth developed by
Segerstrom and Dinopoulos (2007) . First, starting from free trade, we show that a tariff imposition in the South has no effect on the longrun growth rate, but it leads to a temporary increase in the innovation rate and a permanent increase in the imitation rate. Also, it leads to reduced wage inequality between the North and the South and a higher utility for the Southern consumers in the steady state. Second, we show that a tariff imposition in the North has no effect on the longrun growth rate, but it causes a temporary increase in the innovation rate and a permanent decrease in the imitation rate. Also, it leads to enlarging wage inequality between the North and the South for the higher utility provided to Northern consumers in the steady state. Third, we show that a tariff imposition in both regions temporarily raises the innovation rate, but does not change the longrun rate. However, it has little effects on imitation rate and wage inequality.

KEYWORD
NorthSouth Trade , Economic Growth , Tariff , Subsidy

Ⅰ. Introduction
Despite the current trend of globalization, tariffs are still one of the most important trade policy measures for both developed and developing countries.1 Since the establishment of the WTO (World Trade Organization), tariff rates have been reduced throughout the world. Have these tariff reductions accelerated economic growth? How does the effect differ between developed countries and developing countries? How do these tariffs affect economic growth driven by technological progress (an engine of growth)? How do these tariffs affect the wage differential between developing and developed countries? The main objective of this paper is to answer these questions.
RiveraBatiz and Romer (1991a) and Dinopoulos and Segerstrom (1999) studied trade policy in a model with two perfectly symmetric countries that impose the same tariff rate on all imported goods. RiveraBatiz and Romer(1991a) find that, compared with free trade, trade restrictions reduce the global rate of economic growth. Unlike RiveraBatiz and Romer (1991a), Dinopoulos and Segerstrom (1999) supposed two factors (unskilled and skilled labor) that are used in both production and R&D activities and found that trade liberalization not only promotes technological change, it also increased wage inequality within countries.
Segerstrom, Anat and Dinopoulos (1990) introduce a NorthSouth trade model to analyze trade policy issues. In their NorthSouth trade model, Northern firms conduct R&D activities to develop higher quality products. These products are initially discovered and produced by firms in the North (developed countries) and are exported to the South (developing countries). Northern firms that innovate receive patent protection for their products and earn monopoly profits until their patents expire. Then production shifts to the South where wages are lower and the products are exported back to the North. They study the steady state equilibrium effects of prohibitive tariffs designed to protect dying industries from competition with Southern firms. These tariffs lead to higher relative wages for Northern workers and a lower rate of innovation in the North. But their model has "scale effects", namely the property that the steady state rate of economic growth is an increasing function of population size. Jones (1995a) criticizes this scale effect property. A variety of endogenous growth models without scale effects have been developed by Jones (1995b), Kortum (1997), Segerstrom (1998), Young (1998) and Howitt (1999).
Sergerstrom and Dinopoulos (2007) adopt an endogenous growth model without scale effects to develop a new NorthSouth trade model. Nevertheless, they left the analysis of tariff imposition in this model as an open question. We extend the NorthSouth trade model developed by Dinopoulos and Segerstrom (2007) by adding tariffs and subsidies and then analyze trade policy using tariffs. We analyze the steady state effects of tariff imposition. Specifically, we study how the tariff imposition and recycling of its tariff revenue (subsidy to domestic firms) of the North (a developed country) or the South (a developing country) affects the innovation rate in the North, imitation rate in the South, wage differential between the two regions, and longrun growth rate of the world economy. Like Dinopoulos and Segerstrom (2007), the innovation only occurs in the North and the South imitates the Northern products. In this paper, Northern firms devote their resources to innovative R&D that enables them to discover higher quality products and Southern firms also devote their resources to imitative R&D that enables them to copy the stateoftheart quality products. We assume that each government uses its tariff revenue to subsidize domestic R&D. This assumption is based upon the fact that R&D expenditure financed by the government accounts for a substantial percentage in most countries; in particular, 58.6% in Poland, 44.4% in Hungary, and 50.2% in Mexico in 2007 (see Table A1).
We consider the following three cases: first, a tariff imposition only in the South; second, a tariff imposition only in the North; and third, a tariff imposition in both regions. In the first case, we show that a tariff imposition in the South has no effect on the long run growth rate,2 but it leads to a temporary increase in the innovation rate and a permanent increase in the imitation rate. We also show that it reduces wage inequality between the North and the South and leads to higher utility for Southern consumers in the steady state. In the second case, we show that a tariff imposition in the North has no effect on the longrun growth rate, but it causes a temporary increase in the innovation rate and a permanent decrease in the imitation rate. We show that a tariff imposition in the North enlarges wage inequality between the North and the South and leads to higher utility for Northern consumers in the steady state. In the third case, we show that a tariff imposition in both regions temporarily raises the innovation rate, but does not change the longrun rate. The effects on the imitation rate and wage inequality have little effect on the simulation analysis. However, the steady state utility in both regions increased because of quality improvement. The longrun rate of technological change in our model is proportional to the exogenous rate of population growth, and therefore tariffs have no longrun effect, as in other models of economic growth without the scale effect by Jones (1995b) and Kortum (1997).
The rest of the paper is organized as follows. In Section 2, we define our ynamic general equilibrium model of NorthSouth trade with tariffs and subsidies. In Section 3, we identify the steady state conditions for the economy and state the main results. We conclude in Section 4.
1The tradeweighted average tariff for all products of the United States in 2008 was 2%, with 4.1% for agricultural products and 1.9% for nonagricultural products. However, average tariffs vary by industry. In particular, average tariffs were 21.1% for dairy products and 11.4% for clothing. In the case of China, the weightedaverage tariff for all products was 4.3%, with tariffs of 10.3% for agricultural products and 4.0% for non agricultural products. Average tariffs were 27.4% for sugars and confectionery and 16.1% for clothing. 2The longrun growth rate or the steady state innovation rate depends only on the population growth rate n and the R&D difficulty parameter λ, as in Segerstrom (1998). The steady state innovation rate is (n/(1λ)) in this model.
Ⅱ. The Model
We add tariffs and subsidies to the model of Dinopoulos and Segerstrom (2007). Nevertheless, we will follow the same reasoning as in their paper to derive conditions for steadystate equilibrium.3 There are two regions in the world economy, the North (a developed country) and the South (a developing country). Innovation in the North and imitation in the South take place in the context of trade relations between the two regions. Workers in the North are capable of conducting both innovative and imitative R&D, but workers in the South can only conduct imitative R&D because of the insufficiency of highly trained and specialized labor. In the South, technological progress, or imitation, occurs through the import of product designs and production methods developed in the North.
1. Industry Structure and Quality Improvement
There is a continuum of industries indexed by
θ ∈[0,1] in both the North and the South. In all industriesθ ∈[0,1], a firm produces its product of a quality levelj ∈{0,1,2,...} which changes over time as follows: at time 0, the highest qualityj =0, and at time t, the quality can reach the levelj +1 if the highest quality at timet 1 is j and the firm invests in R&D activity.2. Consumers
In both the North and the South, there is a fixed number of households at each time. Each member of a household lives forever and is endowed with one unit of labor. The size of each household grows exponentially at the fixed rate
n >0. Thus, the number of consumers at time t is given bye^{nt} . At time zero, the North haspopulation and the South has
population. The North and the South have the same population growth rate n. Therefore at time t, the supply of labor in the North is
and the supply of labor in the South is
The total supply of labor in the North and the South combined at time
t isConsumers have identical preferences in the North and the South. Temporal preferences of consumers are given by
where
d (j,θ,t ) denotes the quality j consumption of productθ at time t,σ > 1 is the constant elasticity of substitution between products across industries, and δ >1 is the quality parameter. Becauseδ^{j} is increasing in j, consumers prefer the higher quality product.Lifetime utility is given by the discounted sum of the temporal utilities as follows:
where
ρ >n is the constant subjective discount rate.Each consumer maximizes the lifetime utility by choosing quantities of demand of all types and all qualities. This problem is solved in three steps.
The first step is to solve an industry wide static optimization problem for each industry
θ and each time t,where
p (j,θ,t ) is the price ofθ good of quality j at time t, andc (θ,t ) is the consumer's expenditure onθ good at time t. The solution for this problem is to buy only the product with the lowest quality adjusted pricep_{j} (θ )/δ^{j} . Assume that when the two products have the same quality adjusted price, the consumer consumes the highest quality product.The second step is to solve the cross industry static optimization problem. For each t,
where
j (θ,t ) is the quality ofθ good chosen at time t (determined in the first step) andd (θ,t ) is the quantity of demand for this good, and c(t) denotes the total expenditure at time t. Solving this problem yields the individual consumer's demand function as follows. Define a quality measure at time byq (θ,t ) =δ ^{(σ1)j(θ,t)}. Note that sincej (θ,t ) does not depend onc (θ,t ), thenj (θ,t ) does not depend onc (t ) and soq (θ,t ) does not depend onc (t ).The optimal consumption of
θ good of qualityj (θ,t ) is given byThe total spending on
θ good at time t is given byThe third step is the intertemporal expenditure choice problem. Formally:
where
A (t ) is the asset value at time t,w (t ) is the wage rate, andr (t ) is the market interest rate. The solution for this optimal control problem satisfiesHence, in a steady state equilibrium, where the expenditure of an individual consumer,
c (t ), is constant over time, the market interest rate r is equal to the discount rate; that is,r (t ) =p for each t.3. Production Technology and Market Structure
Any firm can exit the industry at any time in the North and in the South. Northern firms enter an industry in the North by discovering higher quality products and Southern firms enter by imitating the stateoftheart products.
There is only one input good, labor, and production technologies of all industries are identical and linear in each region. Northern producers have higher labor productivity than Southern producers. Let
h > 1 be the labor productivity in the North and 1, the labor productivity in the South. Then when labor markets are competitive in each region, the constant marginal cost of a Northern firm equalsw_{N}/h and the constant marginal cost of a Southern firm equalsw_{S} , wherew_{N} andw_{S} are wage rates in the North and the South, respectively. Exporting a product from one region to another incurs a simple form of trade cost. In order to export 1 unit, the exporting firm should produceφ ≥ 1 units.There is an identical production technology across the North and the South, across products, and across quality levels. Goods are produced using one input good, labor, and their production technology exhibits constant returns to scale,
f (l ) =l . This means that one unit of labor produces one unit of output independent of its quality level or geographic location.Labor markets are segmented in the North and the South and are perfectly competitive in each region. The wage rate in the North is denoted by
w_{N} and wage rate of the South is denoted byw_{S} . Each firm in the North has a constant marginal cost equal tow_{N} and each firm in the South has a constant marginal cost equal tow_{S} . We restrict our attention to the case whenw_{N} >w_{S} >w_{N}/δ . The first part (w_{N} >w_{S} ) implies that production shifts from the North to the South when a Southern firm imitates. The second (w_{S} >w_{N}/δ ) implies that the production shifts back to the North when a Northern firm innovates.Governments of the North and the South can impose tariffs on imported goods. The tariff rate of the Northern government is denoted by
τ_{N} and the tariff rate of the Southern government is denoted byτ_{S} .In any industry, firms (the North or the South) producing different qualities are under the Bertrand price competition. In the case of drastic innovation, the new quality leader immediately charges the unconstrained monopoly price and in the case of nondrastic innovation, the new quality leader charges a limited price initially and immediately reverts to charging the unconstrained monopoly price when he learns that the previous quality leader exited the market. We assume a drastic innovation as in Segerstrom and Dinopoulos (2007).
1) Price Determination of Northern Firms
If a Northern firm wins the innovative R&D race, then given the advalorem tariff
τ_{S} in the Southern market, its profit is given bywhere
p_{N} is the price of the Northern product in the North andp ^{*}_{N} is the “posttariff” price of the Northern product in the South,d_{N} is the per capita Northern demand,d_{S} , is the per capita Southern demand. The first part of the Northern firm's profit is from the Northern market and second part is from the Southern market.In the Northern market, since the price elasticity of demand for all
θ isσ and marginal cost of Northern firms isw_{N}/h , the monopoly markup price of Northern firms isIn the Southern market, since Northern firms have to pay a tariff to the foreign (Southern) government when they export the stateoftheart product to the Southern market, they determine a "pretariff" price to maximize the "posttariff" revenue. Then the Northern firm's profit maximization problem is given as follows: for any
τ_{S} where
p is the price level andq is the quantity level. The first order condition isThis is the markup condition
where
is the price elasticity of demand. In this paper, consumers have a constant elasticity of demand
ϵ =σ , and constant marginal cost isw_{N}/h . So the "pretariff" price set by the Northern firm in the Southern market isThis “pretariff” price is the price which the Southern consumer actually confronts. Thus by (7), “posttariff” price
p ^{*}_{N} equalsp_{N} .4 This “posttariff” price is the price which the Northern firm actually confronts after the imposition of tariffs.2) Price Determination of Southern Fir
If a Southern firm wins the imitative R&D race and the North imposes the advalorem tariff
τ_{N} for the Southern product, its profit is given bywhere
p_{S} is the price set by the Southern firm in the Southern market,p ^{*}_{S} is the "posttariff" price set by the Southern firm in the Northern market.The Southern firm sets the monopoly price in the Southern market; that is,
And the “pretariff” price set by the Southern firm in the Northern market is
Thus by (9), the “posttariff” price
p ^{*}_{S} equals top_{S} .5Before deriving an alternative expression for the value of the monopoly profit, we introduce some additional notation. Let
c_{N} (t ) denote the consumption expenditure of the representative Northern consumer at time t and letc_{S} (t ) denote the consumption expenditure of the representative Southern consumer at time t. Then the global consumption expenditure isWhere
is the global per capita consumption expenditure. In the presence of tariffs, Northern consumers face different prices than Southern consumers and we need to take this into account. Let the Northern quality adjusted price index be defined by
and let the Southern price index be defined by
where stars denote exports, subscripts denote production location, is the set of industries with Northern production and is the set of industries with Southern production. Using (3), the Northern consumers demand for a domestically produced good is
The Northern consumer's demand for an imported good (exported by the South) is
The Southern consumer's demand for a domestically produced good is
And the Southern consumer's demand for an imported good (exported by the North) is
Additionally, we find the following relationship:
Using the above mentioned notation, a Northern quality leader in industry
θ at time t earns the following monopoly profit: whereIf a Southern quality leader wins imitative R&D, then its profit is given by
4. Innovative R&D and Imitative R&D
Labor is the only factor of production used by firms that engage in either innovative or imitative R&D activities. When Northern firm i hires
l_{i} labor for innovative R&D, its probability of successI_{i} (in discovering the next higher quality) is given bywhere
γ > 0 indicates the innovative R&D productivity parameter. Thus, as the quality improves, the probability of success goes down.6Southern firms use labor for imitative R&D. When Southern firm
i hiresl_{i} labor for imitative R&D, its probability of success of imitative R&D (imitating the stateoftheart product in industryθ ) is given bywhere
β > 0 is the imitative R&D productivity parameter. As the quality improves, the probability of success goes down. The level ofβ can also be interpreted as the degree of intellectual property rights enforcement.The returns to both innovative and imitative R&D's are assumed to be independently distributed across firms, across industries and across time. The probability that some Northern firms innovate in an industry is given by
The probability that some Southern firms imitate in an industry is given by
At time t, a measure of
m_{N} industries denotes Northern quality leaders, a measure ofm_{S} industries denotes Southern quality leaders, and their sum equals 1, that isIn the steady state, the flow into the industries must be in equal size to the flow out of the industries. If a Southern firm imitates a product of another Southern firm, the profits of both Southern firms become zero because they are involved in the Bertrand price competition. Hence Southern firms imitate the products of Northern firms. If a Southern firm successfully imitates a Northern product, then the production base shifts to the South.7 If a Northern firm successfully innovates a Northern product, this does not cause any shift in the product base. If a Northern firm successfully innovates a Southern product, the production base shifts to the North. Therefore, we obtain the following equation in the steady state.
Using (19), we can derive
m_{N} andm_{S} as a function of the innovation rate and the imitation rate as follows5. R&D Optimization
All firms maximize expected discounted profits and there is free entry to innovative R&D races in the North. Since all firms have the same innovative R&D technology, a Northern quality leader does not engage in R&D activities. Other firms except the Northern quality leader have incentives to engage themselves in innovative R&D races.
In the steady state, the expected benefit from the innovative R&D is equal to the expected cost. And we assume that the government revenue from the Northern tariff is used to subsidize Northern firms. Let
υ_{I} (θ,t ) be the expected discounted profit from innovative R&D ands_{I} (τ_{N} ) the innovative R&D subsidy rate in the North. The subsidy rate depends onτ_{N} andFirm
i ’s probability of successful innovation during time intervaldt is measured byI_{i}dt . Therefore, the expected benefit from innovative R&D is given byυ_{I} (θ,t )I_{i}dt . Expected cost from the innovative R&D during intervaldt is (1 –s_{I} (τ_{N} ))w_{n}I_{i}dt , which equals (1 –s_{I} (τ_{N} ))w_{n}γq (θ,t )I_{i}dt by (17).Thus, we obtain the following steady state R&D optimization condition:
There is also free entry into all imitative R&D races in the South. In the steady state, the expected benefit from the imitative R&D will be equal to the expected cost. Let
υ_{c} (θ,t ) be the expected discounted profits for imitative R&D ands_{C} (τ_{S} ) the imitative R&D subsidy rate in the South. The subsidy rate depends on the Southern tariff rate andFirm
i 's probability of successful imitation during the time intervaldt isC_{i}dt . Thus the expected benefit from imitative R&D isυ_{C} (θ,t )C_{i}dt . Likewise, the expected cost from imitative R&D is (1 –s_{C} (τ_{S} ))w_{C}I_{i}dt which equals (1 –s_{C} (τ_{S} ))w_{S}βq (θ,t )C_{i}dt by (18). Thus, the steady state R&D optimization condition of the Southern firm under the free entry condition is given by6. The Stock Market
1) Northern Firms
In the stock market, households diversify the risk of holding stocks issued by Northern and Southern firms. Through the noarbitrage condition, the return from holding the stock of a Northern quality leader must be the same as the return from an equal sized investment in a riskless bond; that is
where
is the dividend rate from the stock of a Northern quality leader,
is the capital gains rate. By the consumer optimization condition, the market interest rate
r equals the subjective discount ratep in the steady state. Sinceυ_{I} (θ,t ) is constant in the steady state,8 the steady state equilibrium reward from innovative R&D is given by2) Southern Firms
The stock market value of the Southern quality leader can be similarly derived. The return from holding the stock of a Southern quality leader must be the same as the return from an equal sized investment in a riskless bond. Then, the corresponding no arbitrage condition is
where
is the dividend rate from the stock and
is the capital gains rate. The profits of the Southern quality leader are discounted by the market interest rate. The Southern firm has a possibility to lose its business (i.e. exit the industry) by probability I. The steady state equilibrium reward from imitative R&D is given by
7. Steady State R&D Conditions
1) Northern Firms
Using (21) and (24), we get
Substituting π_{N} of (15) into (27) and using the above notation, we rewrite the steady state R&D condition as follows.
The left hand side of (28) is related with the benefit of innovative R&D. The benefit is increasing in
c_{N} andc_{S} , which means that average consumer buys more asy_{N} (t ) ory^{*}_{N} (t ) increase. It is also increasing in initial population capacitiesHowever, the benefit decreases when discount rate ρ, imitation rate of Southern firms
C , or innovation rate of Northern firmsI increases. If the tariff rate of the South, τ_{S}, increases, the benefit of innovative R&D decreases. The cost of innovating increases whenγ increases (innovating becomes more difficult), but it decreases as the R&D subsidy rates_{I} (τ_{N} )increases.2) Southern Firms
Using (22) and (26), we get
Substituting π_{S} of (16) into the (29) and using the above notation, we rewrite the steady state R&D condition as follows.
The left hand side of (30) is related with the benefit of imitative R&D. The benefit is increasing in
c_{N} andc_{S} . It is also increasing in initial population capacitiesHowever, the benefit decreases when the future discount rate ρ or the innovation rate of Northern firms
I increases. And the benefit of imitative R&D is decreasing in Northern tariffs τ_{N}. The right hand side is related to the cost of imitative R&D. The cost increases as the level ofβ increases, but it decreases as the R&D subsidy rates_{C} (τ_{S} ) increases.8. Quality Dynamics
The average quality of products at time t is given by
where λ = δ^{(σ–1)} > 1. When innovation occurs in industry
θ , the quality improves from λ^{J(θ,t)} to λ^{J(θ,t) + 1}. Since the innovation rateI is constant across industries and over time, the quality improvement rate during time interval is given byFrom this equation,
Since the level of relative R&D difficulty
x_{N} =Q (t )/L_{N} (t ) is constant over time and the population growth rate is n, the longrun steady state innovation rate is given byNote that the longrun steady state innovation rate does not depend on the tariff rate, even if the innovation level depends on the tariff rate.
The average quality of all products can be decomposed into the average quality of the Northern leading firms and the average quality of the Southern leading firms as follows:
where
Q_{N} (t ) is the measure of product quality of the Northern firms, andQ_{S} (t ) is the measure of product quality of the Southern firms. The two quality measuresQ_{N} (t ) andQ_{S} (t ) evolve over time in the steady state equilibrium. The quality of the Southern leading firms improves after successful imitations and decreases after successful innovations by Northern firms. Then the time derivative ofQ_{S} (t ) is given byThe quality of the Northern leading firms improves after successful innovations, but decreases after successful imitations. Then the time derivative of
Q_{N} (t ) is given byThe growth rates of
Q_{N} (t ) andQ_{S} (t ) in the steady state are identical. Therefore, using (35) and (36), we obtain9. Labor Markets
1) The Northern Labor Market
We assume that workers can move freely and instantaneously across firms in the North and the South. In each region, we assume the full employment condition and wage rate is determined by supply and demand in each labor market. The Northern labor demand is composed of two parts, manufacturing employment and R&D employment. Since one unit of product requires one unit of labor, the labor requirement of Northern manufacturing companies equals the world demand for Northern products. Thus the production employment in Northern industry
θ is given bySince there are leading industries in the North, the total manufacturing labor demand by Northern firms is given by
Northern R&D employment of industry
θ at timet is Σ_{i}l_{i} =γIq (θ,t ). Therefore, the total Northern R&D employment is given byThe labor supply of the North in time
t isL_{N} (t ). From (40) and (41), the full employment condition in the North isThe Northern manufacturing labor demand increases as
Y_{N} (demand for Northern products by average consumers) increases and(the ratio of Northern products) increases, but decreases as the Southern tariff rate increases. The Northern R&D labor demand is increasing in innovation rare
I and innovative R&D difficultyx_{N} .2) The Southern Labor Market
Similarly, manufacturing employment in Southern industry
θ is given byThe total manufacturing employment in the South is given by
Since the Southern R&D employment in industry
θ at time t is Σ_{i}l_{i} =βCIq (θ,t )dθ , total Southern R&D employment isAnd the labor supply of the South at time t is
L_{S} (t ). From (44) and (45), the full employment condition in the South is given byThe Southern manufacturing labor demand increases as
Y_{S} (demand for Southern products by average consumers) increases and(the ratio of Northern product) increases, but decreases as the Northern tariff increases.
3Specifically, sections 2.1, 2.2, 2.4 follow the same equations as Segerstrom and Dinopoulos (2007). 4. 5. 6This application removes the scale effect of the growth theory criticized by Jones (1995). 7This is called Vernon (1966)’s product cycle model. 8In the steady state condition,
Ⅲ. The Steady State Equilibrium
1. Existence of the Steady State Condition
We determine the Northern steady state condition from the equations (28), (42) and
x_{N} =Q (t )/L_{N} (t ).Similarly, the Southern steady state condition from the equations (30), (46) and
x_{N} =Q (t )/L_{N} (t ) yields the following equationNote that tariff imposition in the South does not affect the Northern steady state condition. It only affects the Southern steady state condition through the subsidy for Southern firms. Also note that tariff imposition in the North does not affect the Southern steady state condition. It only affects the Northern steady state condition through the subsidy for Northern firms.
From the two steady state conditions in the two regions, we identify the imitation rate
C , the shortrun innovation rate in the steady state, and the time path ofQ (t ) usingThus the quality improves (technology progress) faster as the shortrun innovation rate gets higher.
To get the formula for the steady state wage differential
w =w_{N}/w_{S} , we use (28) and (30) to getUsing (11), (13),
Thus from (50), we can derive the following wage differential equation:
This wage differential equation implies that the NorthSouth wage gap w is indirectly related to the change in tariffs through the change of (ρ +
I )/(ρ +I +C ), which is the reward for innovative R&D relative to the reward for imitative R&D. Clearly, it is also directly related to tariffs, τ_{N} and τ_{S}.2. Effects of Tariffs
To analyze the effects of a tariff imposition, we compare the equilibrium without a tariff with the equilibrium in the following three cases: first, when only the South imposes a tariff; second, when only the North imposes a tariff; and third, when both regions impose tariffs. From (47) and (48), we obtain the following steady state conditions without a tariff:
1) Effects of Tariff Imposition in the South
Assume, that there is no tariff in the North and that the South imposes tariff τ_{S} on imported Northern stateoftheart products. Assume that the Southern tariff revenue subsidizes domestic imitative R&D. Recall that the tariff imposition does not change the Northern steady state condition and so (52) prevails in the North. The Southern steady state condition shifts from (53) to (48) (see Figure 2).
In Figure 2, the steady state equilibrium without a tariff is denoted by
E. After tariff imposition in the South, the decreasing curve indicating the Southern steady state condition shifts to the right. Thus we obtain the new steady state equilibrium atE ^{*}, which has the higher shortrun innovation ratex_{N} and the higher imitation rateC . It follows from (49) that the average quality of products Q improves asx_{N} increases. And it follows from (20) that the ratio of the Southern quality leaderm_{S} increases and the ratio of the Northern quality leaderm_{N} decreases asC increases. This means that more industries shift from the North to the South. It follows from (51) that wage differential w decreases as bothC and τ_{S} increase. It should be noted that all these effects are due to Southern imitative R&D subsidies financed by tariff revenue. As is clear from (48), if the tariff revenue is not used for subsidizing domestic R&D, there is no steady state effect of Southern tariff imposition. Therefore, we obtain the following result.Proposition 1 If the South imposes a tariff on Northern innovative products, then it leads to a higher imitation rate and a higher shortrun innovation rate (C↑ and x_{N}↑) and the steady state relative wage decreases (w↓), the longrun innovation rate (I=(n/(λ1))) does not change, and industries shift from the North to the South (m_{N}↓, m_{S}↑). The steady state of Southern tariff imposition is quite intuitive. The subsidy to the Southern firm from tariff revenue makes imitation R&D more efficient. This leads to more imitation in the South and more industries will move from the North to the South (m_{N}↓, m_{S}↑).
The profit of Northern firm (π_{N}) will decrease caused by the decrease in demand (See 15). The decrease of the profit of Northern firms reduces the R&D benefit of innovation (R&D Reducing Effect) (See 28). On the other hand, the decrease of demand in Northern products leads to a decrease in labor demand in the manufacturing sector in the North (See 39). More Northern workers become available for employment in the Northern R&D sector which makes it more attractive for Northern firms to expand their R&D activities (R&D Enhancing Effect). These two effects offset the Northern steady state equilibrium.
But the industry shift from the North to the South because of the Southern subsidy makes more R&D labor available in the North. Therefore, the shortrun innovation rate will jump and technological change will accelerate, but the industry level innovation rate will gradually fall back to the original steady state level
I =n /(λ – 1), as R&D becomes relatively more difficult.2) Effects of Tariff Imposition in the North
Assume that there is no tariff in the South and that the North imposes tariff τ_{N} on imported Southern products. Assume that the Northern tariff revenue subsidizes domestic innovative R&D. Recall that the tariff imposition does not change the Southern steady state condition and so (53) prevails in the South. The Northern steady state condition shifts from (52) to (47) (see Figure 3).
After the Northern tariff imposition, the Northern steady state condition curve shifts to the right and the original steady state equilibrium E moves down to a new steady state equilibrium
E ^{*} in Figure 3. Therefore, the shortrun innovation ratex_{N} increases, but imitation rate C decreases. From (49), the average quality of products Q improves. From (20), the ratio of the Southern quality leaderm_{S} decreases and the ratio of the Northern quality leaderm_{N} increases. This means that more industries shift from the South to the North. From (51), wage differential w increases. Therefore, we obtain the following result.Proposition 2 If the North imposes tariffs on Southern imitative products, then the imitation rate decreases (C↓), the shortrun innovation rate increases (x_{N} ↑), the steady state relative wage increases (w↑), the longrun innovation rate (I=(n/(λ1))) does not change, and industries shift from the South to the North (↑,↓). The steady state of Northern tariff imposition is also intuitive. The subsidy to the Northern firm from tariff revenue makes innovation R&D more efficient. This leads to more innovation R&D in the North. Therefore, the shortrun innovation rate will jump and technological change to accelerate, but the industry level innovation rate will gradually fall back to the original steady state level
I =n /(λ – 1) as R&D becomes relatively more difficult.The profit of Southern firm π_{S} will decrease because of the decrease in demand (See 16). The decrease of the profit of Southern firms reduces the R&D benefit of imitation (R&D Reducing Effect) (See 30). So the imitation rate (C ↓) will decrease. Therefore, more industry will move from the South to the North (
m_{N} ↓,m_{S} ↑)On the other hand, the more industry shifts from the South to the North, there will be a decrease in labor demand in the manufacturing sector in the South (See 43). More Southern workers become available for employment in the Southern R&D sector. This makes it more attractive for Southern firms to expand their R&D activities (R&D Enhancing Effect). These two effects offset the Southern steady state equilibrium.
3) Effects of Simultaneous Tariff Imposition in the North and in the South
Now assume that both the North and the South impose tariffs on imported goods and subsidize their domestic R&D programs. At that time, both the Northern steady state condition and the Southern steady state condition will change to (47) and (48) (see Figure 4).
In this case, both the Northern steady state curve and the Southern steady state curve shift to the right. The original steady state equilibrium E moves to a new steady state equilibrium
E ^{*} as depicted in Figure 4. The shortrun innovation rate increases, but the effects on the imitation rate of the South and wage differential are not determined analytically. However, we expect there will be no big change in the imitation rate of the South or wage differential. The welfare of both regions can not be analytically determined. Therefore, these questions need numerical analysis. Eventually, we obtain the following proposition.Proposition 3 If both the North and the South impose tariffs, then the shortrun innovation rate increases and the longrun innovation rate (I=n/(λ1)) does not change. 3. Steady State Utility Paths
Let
V_{N} denote the total market value of all Northern firms att = 0 and letV_{S} denote the total market value of all Southern firms att = 0. To solve the model, these market values are in the steady state equilibrium.First, consider how the price indices evolve over time. Using (37), we obtain that
Thus the Northern price index
P_{N} (t ) decreases over time with product qualityQ (t ) and increases with the Northern tariff rate τ_{N}. The same holds for the Southern price index,Thus the Southern price index
P_{S} (t ) decreases with product qualityQ (t ) and increases with the Southern tariff rate τ_{S}.During the lifetime of the firm,
q (θ,t ) is constant,p_{N} =w_{N}/αh is constant sincew_{N} is constant,P_{N} (t )^{1 – σ}/Q (t ) andP_{S} (t )^{1 – σ}/Q (t ) are constants, andQ (t )/Q (t ) = (λ – 1)I =n , soL_{N} (t )P_{N} (t )^{1 – σ} andL_{S} (t )P_{S} (t )^{1 – σ} are also constants over time. So the firm's profit flow π_{N}(θ,t ) is also constant over time. Consequently, the market value of a Northern firm does not change over the course of the firm's lifetime. Using this information,Substituting using (35) and
we determine that the market value of all Northern firms at t=0 is
Using similar notations, the market value of all Southern firms at
t = 0 isOther things being equal, the market value of firms in a region is higher when workers earn higher wages and when innovating is relatively more difficult.
Having determined the market value of the firms, let's solve for consumer expenditures. Let
A _{N(t)} andA _{S(t)} denote the financial assets of the representative Northern consumer and Southern consumer. Assume that Northern consumers own the Northern firms and Southern consumers own the Southern firms, that is,A_{N} (t ) =V_{N} (t )/L_{N} (t ) andA_{S} (t ) =V_{S} (t )/L_{S} (t ).The intertemporal budget constraint of the representative Northern consumerA_{N} (t ) =w_{N} + ρA_{N} (t ) –c_{N} –nA_{N} (t ) can be rewritten assince the growth rate of
A_{N} (t ) must be constant over time in any steady state equilibrium. The intertemporal budget constraint implies thatA_{N} must be constant over time in any steady state equilibrium. Using (56), the representative Northern consumer's expenditure at steady state isThe representative Northern consumer's expenditure is wage income plus interest income on financial assets appropriately adjusted to take into account the splitting of the financial assets that result from population growth. Using (57), the representative Northern consumer's expenditure at the steady state is
We now turn to the steady state utility paths of representative consumers in the North and South, respectively. For the typical Northern consumer, the utility at time t is
Substituting (11) and (12) yields
From (58) and (54), the steady state utility of the North is
and the corresponding calculations for the Southern consumer yields
From (59) and (55), the steady state utility of the South is
As Dinoupoulos and Segerstrom (2007) identified that, since the old new steady state equilibrium paths involve the same rate of economic growth, we can compare utility levels at time
t = 0 in the old and new steady states to determine whether the change makes consumers better off in the long run.Ⅳ. Simulation Analysis
The results of the simulation analysis should be interpreted as suggestive rather than conclusive, but these results give some implications on the problems which could not be solved analytically. We supposed that the following benchmark parameter values as in Dinopoulos and Segerstrom (2007) were ρ = 0.07,
n = 0.014σ = 1.5 and
h = 1.67. Additionally λ = δ^{(σ–1)} = 1.3. The results from the computer simulations are reported in Table 1 and the first column shows the endogenous variables for the benchmark parameter values. The second column shows the changes of endogenous variables in the case of a tariff imposition in the South. We supposed the Southern tariff rate is 10% and this tariff revenue was used to subsidize R&D. For analytical simplicity, I supposed that a 10% increase in the tariff rate comes down to the 10% reduction of labor cost. i.e. we supposed that the labor productivity will increase 10% because of the improved productivity due to the subsidy of R&D. The third column shows the changes of the endogenous variables in the case of a tariff imposition in the North. I also supposed that a 10% increase in the Northern tariff rate comes down to the 10% reduction of labor cost. The fourth column shows the changes of the endogenous variables in the case of a tariff imposition in both regions. We supposed that the tariff rate is 10% in both regions and these tariff revenues were used to subsidize R&D in each region.First, the results confirm proposition 1. According to a tariff imposition in the South,
C increased from 0.048534 to 0.055627 andx_{N} slightly increased from 10.804 to 10.87. Also the industry moved from the North to the South. The share of the Northern products (m_{N} ) decreased from 0.49019 to 0.46999 and the share of the Southern products (m_{S} ) increased from 0.50981 to 0.53001. The relative wage (w ) decreased from 2.1689 to 1.8909. The steady state welfare of the Northern consumers decreased (u_{N} (0) fell from 3.9094 to 3.8320) and the steady state welfare of the Southern consumers increased (u_{S} (0) rose from 3.5457 to 3.6343).Second, the results confirm proposition 2. According to a tariff imposition in the North,
C slightly decreased from 0.048534 to 0.044802 andx_{N} increased from 10.804 to 11.304. The industry moved from the South to the North. The share of the Northern products (m_{N} ) increased from 0.49019 to 0.51019 and the share of the Southern products (m_{S} ) decreased from 0.50981 to 0.48981. The relative wage (w ) increased from 2.1689 to 2.4851. The steady state welfare of the Northern consumers increased (u_{N} (0) rose from 3.9094 to 4.0917) and the steady state welfare of the Southern consumers slightly increased (u_{S} (0) rose from 3.5457 to 3.5568). Even through the Southern consumers confront a higher price for the imported Northern products due to the tariff, the Southern consumers receive a higher quality of imported Northern products. So the overall steady state utility of the Southern consumers slightly increased.Third, the results confirm proposition 3 and show more clearly the undetermined effects. According to a tariff imposition in both regions,
C remains almost the same as the benchmark solution, butx_{N} greatly increased from 10.804 to 11.367. The tariff imposition in both regions had little effect on the proportion of the industry in each region. It also had little effect on the relative wage (w slightly changed from 2.1689 to 2.1693). The steady state welfare of the Northern consumers increased (u_{N} (0) rose from 3.9094 to 4.013) and the steady state welfare of the Southern consumers slightly increased (u_{S} (0) rose from 3.5457 to 3.6466). As such, it is because the effect of quality improvement exceeded the effect of the rising prices of imported goods in both regions.Ⅴ. Concluding Remarks
We adopted a dynamic general equilibrium model of NorthSouth trade with scale invariant growth developed by Segerstrom and Dinopoulos (2007) to analyze the steady state effects of tariff imposition. We consider the effects of a tariff starting from a situation of free trade. Assuming that each government uses tariff revenue to subsidize domestic R&D, we showed that tariff imposition in the South leads to more copying of Northern products, faster technological change, lower wage inequality between the North and the South and higher welfare for the Southern consumers. Tariff imposition in the North leads to less copying of Northern products, faster technological change, higher wage inequality between the North and the South and higher welfare for the Northern consumers. Tariff imposition, both in the North and the South, leads to faster technological change and higher welfare in both regions, but has little effect on imitation rate and wage differentials.

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[Figure 1] World Market Structure

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[Figure 2] The Steady State Effects of a Southern tariff and Southern R&D Subsidy

[Figure 3] The Steady State Effects of a Northern Tariff Imposition.

[Figure 4] The Steady State Effects of Northern Tariff and Southern Tariff with Subsidies.

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[Table 1] The Result of Computer Simulations