Non-scale effects of international technological-knowledge diffusion with
human capital on trade, on inter-country gaps and on wages
Óscar Afonso* and Ana Maria Bandeira†
Abstract
By connecting the North-South diffusion and the bias of non-scale technological knowledge and by
considering endogenous human capital, this paper relates the technological-knowledge diffusion with
levels, inter-country (technological-knowledge and human-capital) gaps, growth rates, wage-inequality
paths and specialisation patterns. Inter-country gaps fall towards the steady state and thus the South
produces more final goods at the end of the adjustment process. Moreover, it exports relatively more
final goods of the type that use more intensively the relatively abundant human capital and imitated
intermediate goods. However, outputs, wages and prices remain different and differences in prices
originate the intra-country wage-inequality paths observed in developed and developing countries.
Keywords: Technological knowledge; International trade; Human capital; Wage inequality.
JEL classification: F43, J31, O3.
August 2010
*
CEFUP, OBEGEF, Faculdade de Economia, Universidade do Porto.
Please address correspondence to Oscar Afonso (oafonso@fep.up.pt), Faculdade de Economia, Universidade do
Porto, Rua Roberto Frias, 4200-464 Porto, Portugal; Phone: +351225571100; Fax: +351225505050.
†
Instituto Superior de Contabilidade e Administração do Porto.
1
1. Introduction
Overall, empirical evidence detects, in developed (North) and developing (South) countries,
strong technological-knowledge progress, enlarged international-trade flows, increase in wage
inequality in favour of skilled labour and rise in the skilled-labour share, since the early 1980s
(e.g., Coe and Helpman, 1995; Machin and Van Reenen, 1998; Berman et al., 1998;
Acemoglu, 2003; Robertson, 2004; Zhu and Trefler, 2005; Avalos and Savvides, 2006).
We analyse mechanisms through which North-South technological-knowledge diffusion
by imitative R&D under trade affects the specialisation patterns and thus trade, skilled and
unskilled human-capital accumulation, non-scale skill-biased technological-knowledge
progress, economic growth and wage inequality. We follow and contribute to two main lines
of research: technological-knowledge diffusion growth models (e.g., Grossman and Helpman,
1991); and wage-inequality growth models (e.g., Acemoglu and Zilibotti, 2001).
Neither the technological-knowledge diffusion nor the wage-inequality literature attests
all the above trends. The former ignores wage-inequality analysis. The latter stresses labour
levels and comprises two rival approaches (e.g., Zeira, 2007): (i) the trade approach, anchored
in the Stolper-Samuelson theorem (e.g., Leamer, 1998; Wood, 1995a, b), predicts, however, a
rise in relative unskilled wage in developing unskilled abundant countries;1 (ii) the skillbiased technological change (SBTC) approach, rooted in the market-size effect on the
technological-knowledge bias that drives wages (e.g., Acemoglu, 2002), predicts, in turn, a
rise in relative unskilled technological knowledge and thus in relative unskilled wage in
developed skilled abundant countries due to enlarged trade with developing ones.
In the SBTC approach, a larger skilled labour level creates a larger demand for R&D
directed towards improvements in inputs used in goods produced by skilled labour, thus
1
Accordingly, recent theoretical and empirical studies try to rethink the trade effects on wages (e.g., Krugman,
2008; Verhoogen, 2008; Broda and Romalis, 2009; Burstein and Vogel, 2009; Egger and Kreickemeier, 2009).
increasing relative skilled wages. However, the direction of R&D is also affected by prices,
since more expensive goods generate higher profits for producers. The relative skilled labour
abundance increases the competitive price of goods produced by unskilled labour and, thus,
the demand for R&D directed towards advances in goods produced by unskilled labour. When
the developed country exports inputs including its R&D results, it benefits from the higher
prices of goods produced by skilled labour in the developing country. The profit opportunities
redirect R&D towards inputs that increase the marginal productivity and thus skilled wages.
Instead of fixed labour levels, we consider endogenous human-capital production; thus,
growth engine is the accumulation of both technological knowledge and human capital. To
focus on the price-channel mechanism, we remove scale effects in line with the dominate
literature on scale effects (e.g., Jones, 1995a, b). And considering international technologicalknowledge diffusion, through trade, we connect the two major wage-inequality approaches, as
is empirically supported by, e.g., Jaumotte et al. (2009). Indeed, considering human-capital
accumulation, the price channel and technological-knowledge diffusion, we propose a model
capable of generating predictions compatible with all trends described above.
The North is more productive due to better institutions, innovative R&D (e.g., Aghion
and Howitt, 1992) and higher human capital. South R&D results are imitations of innovations
and it has a marginal cost advantage in production (e.g., Grossman and Helpman, 1991).
Thus, the South imports intermediate goods, where R&D is applied, that have not yet been
imitated and exports those previously imitated.2 There is complementarity between inputs,
human capital and intermediate goods, and substitutability between two technologies in
competitive final goods production (e.g., Acemoglu and Zilibotti, 2001; Afonso, 2008).
Goods are internationally tradable and intermediate goods, by including R&D results,
are the diffusion vehicle. Trade, which starts when the North is in its pre-trade steady state, is
2
Thus, countries have access to the same technological knowledge, by either domestic production or imports.
3
balanced; i.e., in trade-theory tradition, there are no North-South movements of assets (human
capital, institutions and shares of firms that have the exclusive use of designs) and thus
interest rates and wages are domestically found. As the pattern of final-goods specialisation is
decided by the endogenous relative productivity of each country in each good, which, in turn,
is affected by the endogenous human-capital levels, the dynamic comparative advantage is
explained by a dynamic version of both the Ricardian and the Heckscher-Ohlian models.
Following Mincer (1993) and Lucas (1993), the time spent accumulating human-capital
is split between school and on-the-job-training (OJT). Relative to skilled, unskilled human
capital is OJT intensive and is less productive. Maoz and Moav (2004), e.g., in a two-period
model show that human-capital acquisition has a negative effect on the skill premium. In our
continuous-time model, the skill-premium per worker (i.e., the relative wage of workers who
accumulate skilled human capital) can rise even when the skill-premium per unit of human
capital (i.e., the relative wage of skilled human capital) drops, due to the relative rise of
skilled human capital per worker. Moreover, human capital and Northern technological
knowledge are crucial for growth, R&D bias and wage paths.
By relating the diffusion and the bias of technological knowledge and by considering
endogenous human capital, we relate, through the (final-goods) price channel mechanism, the
technological-knowledge diffusion with levels, inter-country technological-knowledge and
human-capital gaps, growth rates, wage paths and specialisation patterns. We analyse the
arising Southern level effects, steady-state effects and transitional dynamics, which reveals a
convergent process between countries (described by the decrease in inter-country gaps).
Differences in human-capital levels impose different prices, which, in turn, increase the
relative demand for skilled-specific new designs thus boosting the relative worldwide supply
of skilled-specific intermediate goods. In particular, as the North-South average relative price
of final goods produced by skilled human capital (the one that is relevant under trade) is
4
always higher than the one existing in pre-trade North, thus, by the price channel,3 trade
relatively redirects technological knowledge in favour of intermediate goods used with skilled
human capital, which relatively boosts wages of skilled workers in both countries.
This originates more moderate paths for technological-knowledge bias and intra-country
wage inequality: e.g., the initial Southern level effect is reverted and the wage-inequality path
is milder than in pre-trade North: a dynamic equivalent to the static Stolper-Samuelson result
is found and a temporary partial dynamic equivalent to the static Rybczynski theorem occurs.
Level and growth effects benefit the South. However, due to the exogenous-productivity
and human-capital gaps, inter-country steady state output and wages levels stay different but
they grow at the same rate: a dynamic Schumpeterian factor-price equalisation result arises.
Both countries produce, consume and export both types of final goods since they always
have both human-capital types. The South: (i) produces more final goods at the end of the
adjustment process since it becomes more developed – inter-country gaps fall; (ii) exports
relatively more final goods of the type that use more intensively the relatively abundant
human capital – directly in line with the Hechscher-Ohlian model; (iii) produces, consumes
and exports imitated intermediate goods – directly in line with the Ricardian model.
In section 2, the paper proceeds to characterise both economies and the international
market. In section 3 we derive the dynamic general equilibrium, we obtain the level, steadystate and transitional-dynamics effects, and we analyse the comparative statics and dynamics
resulting from alternative parameters. In section 4 we present some concluding remarks.
2. Economic structure
By expanding the closed-economy endogenous R&D-growth model with fixed labour levels
in Afonso (2008), we define the productive setup, which, excluding some parameter values
and R&D activities, is common to both countries. Each economy is populated by infinitely-
3
The path of prices is empirically supported by studies, such as Krueger (1997) and Broda and Romalis (2009).
5
lived individuals and population growth is zero. Individuals choose between consumption and
savings on income allocation, and between production and human-capital accumulation on
time allocation. Competitive final goods use unskilled, L, or skilled, H, human capital with Lor H-specific quality-adjusted intermediate goods, which are produced under monopolistic
competition by joining units of aggregate output and designs (e.g., Aghion and Howitt, 1992).
Designs are obtained through innovative and imitative R&D.
2.1. Individuals
A time-invariant number of heterogeneous individuals decide income and time allocation.
Income is partly spent on consumption, C, of aggregate final output, Y, and partly lent in
return for future interest. Time, t, is divided between accumulation of human capital, and
working to earn a share of Y proportional to the individual’s human capital. Heterogeneity is
arrested by the accumulated human-capital type, m = L, H.
 C( t )1 θ 1
1
The lifetime utility function 0
exp(   t ) dt , where  >0 is the subjective discount
rate and  >0 is the relative risk aversion, states individuals (identical) preferences. Savings
are the accumulation of financial assets, K, which have return – the interest rate, r, that, due to
arbitrage in the domestic assets markets, only relies on t. Lending takes the form of ownership
of profitable firms (the ones producing intermediate goods). The value of these firms, in turn,
is the value of patents. The budget constraint equalises savings plus consumption to income
earned at t: K (t )  C (t )  r (t ) K (t )   1  u F ,m (t )  uT ,m (t ) w(t )m(t ) , where wm(t) is the wage per unit
m L , H
of m at t; uF,m(t) and uT,m (t) are the fractions of t spend by m, respectively, at school and OJT –
thus, OJT is costly, in the sense that it needs time away from work (e.g., Mincer, 1993).
Individuals accumulate either H or L using schooling and OJT. As in Lucas (1988), the
productivity of the time spent increases with the individual’s human capital. A constant
elasticity of substitution (CES) accumulation function is considered
6


 



m ( t )  



  m  F uF, m ( t )  ( 1 m )  T uT, m ( t )  
 
m ( t )    

  Schoolm (t ) 
 OJTm (t )  
1/ 
  m , where:
(1)
 m is the depreciation rate of m; terms within square brackets are schooling and OJT inputs;
F and T are efficiency parameters assessing schooling and OJT productivities (we assume
that F  T   m, otherwise m falls).  m  [0, 1] is the intensity parameter, which determines
the relative weight of the two inputs, and we consider that  H >  L, such that H is school
intensive. Assuming that H is more productive, the idea is that schooling provides wide and
flexible human capital, while OJT provides more specific skills (e.g., Mincer, 1993); (1)
allows schooling and OJT to be either complements or substitutes, relying on the substitution
parameter : complements if  < 0 and substitutes if 0 <   1 (e.g., Mincer, 1993).4
Individual maximises lifetime utility, subject to the budget constraint, to (1) and the “no
Ponzi games” condition lim K m (t ) exp (  t )  0 . The result for the consumption path, which is
t 
independent of m, is the standard Euler equation:

r (t )  
, where Cˆ  C .
Cˆ ( t ) 

C
(2)
As to time allocation, the resulting optimal ratio is independent of t, but reliant on m,
uT , m
uF ,m


( 1m ) χT
m χ F

(1 ) 1
. If inputs are substitutes but not perfect, the optimal time-allocation ratio
is positively related to the ratio
F
T
, and vice-versa if they are complements. In this case, for
example, an increase in the time devoted to the higher productivity input requires an increase
in the time allocated to the lower productivity input. An interior solution to the individual
4
Empirical evidence tends to emphasise the complementary nature of schooling and OJT (e.g., Bartel and
Lichtenberg 1987; OECD, 2001; Brunello, 2004). High substitutability, coupled with higher efficiency of
schooling, implies that skills are better obtained at school. In this case, constraining human-capital production to
school tends to be enough, mainly for school intensive H. High complementarity means that schooling is far
from providing all such skills, which supports the claims in growth-human-capital literature that the lack of an
empirically robust relationship is partially caused by the exclusion of OJT (e.g., Lucas, 1993).
7
maximization problem entails positive levels of K and m, which is not sustainable unless their
returns are equalized at all times. The following resulting condition, where the optimal timeallocation ratio is expressed in terms of the given wage dynamics and r, ensures this:
wˆ m (t )  r ( t )  m , where  m  
1/ 
m
 u 
 F 1  T ,m 
 u F ,m 
 1 (1 )
  m and
wˆ m 
w m
wm
(3)
Hence, the optimal time-allocation ratio place limitations on the relation between ŵm
and r, and investments in human-capital accumulation and in K are complementary. If r is
high today, wage dynamics are also high, and this relationship is always balanced.
2.2. Final-goods sector
Each final good, n  [0, 1], is produced by one of two technologies. The unskilled (skilled)
technology uses employed unskilled (skilled) human capital, Lw (Hw), complemented with a
continuum of unskilled (skilled)-specific intermediate goods j  [0, J] (j  [J, 1]).
Both countries – North, N, and South, S – produce final goods using unskilled (skilled)technology, since, at each t, they have L (H) and have access to the top quality intermediate
goods. We assume that in the L (H)-technology, S has a comparative advantage in final goods
indexed by smaller (larger) ns, n[ 0, nL ] ( n[ nH , 1] ). Thus, N has a comparative advantage in
n [ nL , nH ] , thus inducing the pattern of final goods specialisation in Figure 1.5
Figure 1. Pattern of final-goods specialisation
0
________________________ _______________________________ 1
    n L    n    nH    n
Produced by
S
Produced by
N
Produced by
N
Produced by
S

Consumed by both countries
The constant returns to scale production function of n at t is,
5
The ordering index has been built in this way only for analytical convenience: it eases the calculation of n
since only one country produces goods around n ; nL ( nH ) is an endogenous competitive equilibrium threshold
final good that indicates the switch from S (N) L (H)-technology to N (S) L (H)-technology, at each t.
8




J

 
   zn (k , j , t ) 1  d j   AS1 /  n(t ) - n  l L w, n , S (t )   A1N/  n l L w, n, N (t ) 
 



  0
 

 


 
1



  J
Yn ( t )   AN    zn (k , j , t ) 1  d j   n H (t ) - n  l Lw, n , N (t )    zn (k , j , t ) 1  d j  n - n L(t)  h Hw, n , N (t )  ,



 
 
 J
 
  0


  1
 


 
   zn (k , j , t ) 1  d j  AS1 /   n - n(t ) h Hw, n , S (t )   A1N/  1 - n  h Hw, n , N (t ) 
J







 
 
 

(4)
for, respectively, n  [ 0 , n ] , n  [ nL , nH ] and n  [ n , 1] , where the adjustment terms n (t )  n , n,
nH (t )  n , n  nL (t ) , n  n (t ) and 1  n convert the index n into an ordering index of the
comparative advantage of each technology and of each country in production of n.
By assumption, the productivity level A, reliant on domestic institutions, is distinct in
both countries: AS < AN. Considering zn ( k , j, t )  q k ( j, t ) x n ( j, t) , the integrals assess the role of
quality-adjusted intermediate goods to production: 1–, 0<  1, is the intermediate-goods
input share; q > 1 is the size of each quality upgrade; k(j, t) is the top quality rung in j at t,
since only top qualities are used in equilibrium; and x is the domestic or imported quantity.6
The human-capital contribution is evaluated by terms with exponent , which is the
input share. These terms include the employed quantities in n, Hw,n or Lw,n, and thus those that
work and earn a wage.7 An absolute productivity advantage of H over L is captured by h and
l, assuming that h > l  1. A relative productivity advantage of either m-type human capital or
country is captured by adjustment terms, implying, for example, that H is relatively more
productive in final goods indexed by larger ns and that N has a comparative advantage in
n [ nL , nH ] . As we will see later on, the thresholds n , n L and nH are endogenous.
Considering pn and p(j) the prices of n and j, respectively, which are given for the
perfectly competitive producers of final goods, it is straightforward to verify the demand for
the top quality of j by the representative producer of: (i) n  [ 0 , n ] , where j  [0, J], is:
6
S is no longer limited by its own ability to produce such goods; it can import top quality goods that have not yet
been imitated and export top quality goods that it has imitated by under-pricing the innovators in N.
7
mw is all employed m-type human capital, and in (4) mw = Lw if 0 < j  J and mw = Hw if J < j  1.
9
 pn , S (t )

x n, S ( j , t )   n (t )  n  l L w,n, S (t ) 
AS (1 )


p
(
j
,
t
)


 pn, N (t )

x n, N ( j , t )  n l L w, n, N (t ) 
AN (1 )
 p ( j, t )

1/ 
q k ( j, t )
1/ 
q k ( j, t )
 - 1 (1 )
 - 1 (1 )
, if 0  n  nL(t ) ;
, if nL(t )  n  n (t ) ;
(5a)
(5b)
(ii) n  [nL , nH ] , where j  [0, J] for n  [nL , n ] and j  [J, 1] for n] n , nH ] , is:
 pn, N (t )

x n, N ( j , t )   n H (t )  n  l Lw, n, N (t ) 
AN (1 )


p
(
j
,
t
)


1/ 
 pn, N (t )

x n, N ( j , t )   n  n H (t )  h H w, n, N (t ) 
AN (1 )


 p ( j, t )

q k ( j, t )
 - 1 (1 )
, if nL(t )  n  n (t ) ;
q k ( j, t )
 - 1 (1 )
, if n (t )  n  nH (t ) ; (6b)
q k ( j, t )
 - 1 (1 )
, if nH (t )  n  1 ;
(7a)
, if n (t )  n  nH(t ) .
(7b)
1/ 
(6a)
(iii) n  [n , 1] , where j  [J, 1], is:
 pn , S (t )

x n, S ( j , t )   n  n(t ) h Hw,n, S (t ) 
AS (1 )


p
(
j
,
t
)


 pn, N (t )

x n, N ( j , t )   1 n h H w, n, N (t ) 
AN (1 )


 p ( j, t )

1/ 
1/ 
q k ( j, t )
 - 1 (1 )
Producers of final goods demand more intermediate goods when their respective
product prices are high, prices of intermediate goods are low, and their employed human
capital and intermediate goods quality are higher. The output aggregate in each country is:


n
 n
 
 Y   N  exp  n ln Yn , N ( t ) d n  n H ln Yn , N ( t ) d n  
L
1



  , where
Y ( t )   p n ( t ) Yn ( t ) d n  
0

1
 n
 

  S  exp   L ln Yn ,S ( t ) d n  n ln Yn ,S ( t ) d n  
 YS
0
H


 

(8)
(i) the Southern marginal cost, S, is smaller than the Northern, N; i.e., S < N = 1;8 (ii) not
consumed resources, Y-C, are used in intermediate goods production, X, and in R&D, R.
2.3. Intermediate-goods sector
Since, in each country, Y is the input in the production of each j, the marginal cost of
producing j equals  and is independent of its quality and identical across all domestic js. The
production of j requires a start-up R&D cost, either in a new design in N or in its imitation in
S via reverse engineering. This investment is recovered if profits are positive within a certain
period. This is guaranteed by costly (innovative or imitative) R&D and by domestically
enforced patents that protect, intra but not inter-country, the leader firm’s monopoly, at the
8
As stated below, this assumption is crucial under trade of intermediate goods, since competitiveness of the
imitators rests on the assumption that S, the developing country, has a marginal cost advantage (lower wages).
10
same time disseminating acquired knowledge to other domestic firms. Thus, knowledge of
how to produce the top quality is intra-country public (non-rival and excludable) and intercountry semi-public (non-rival and partly excludable).
Assuming balanced trade without international mobility of the other factors and assets,
S can participate in trade by exporting some final and intermediate goods. Indeed, each j is
produced either in N, when embodies the latest innovation, or S, if results from imitation at a
lower cost of the latest innovation. In any case, only top quality intermediate goods are traded.
Thus, each monopolist uses the latest quality, qk, and has access to the entire world market.
Whether it is an innovative or imitated design, it relies on price competition in this market.
S is transmitted to the intermediate goods production, affecting worldwide optimising
limit pricing (e.g., Grossman and Helpman, 1991, Ch. 12). The dynamics of competitive
advantage in j is endogenous and relies on innovations and imitations. The three feasible
sequences of successful R&D outcomes and their limit pricing at t, given k at t-dt, are:9
pm, N , N ( j )  pN , N ( j )  q N , pm, S , N ( j )  pS , N ( j )  q  S and pm, N , S ( j )  pN , S ( j )  N .
(9)
These limit prices stress the market power of the producer of j and are given for finalgood producers. The 1st occurs without trade and is the highest: the entrant at t, in N,
competes with the incumbent at t-dt, in N, at the same , but with better quality (k+1 > k). The
2nd is smaller: the entrant at t, in N, improves quality (k+1 > k), but competes with the
incumbent at t-dt, in S, with lower . The 3rd is also smaller: the entrant at t, in S, with lower
, competes in the same quality rung with the incumbent at t-dt, in N.
9
Without trade, j  [0, 1] in N (S) embodies the latest innovation (imitation) and there are no feedback effects
between N and S. The profit-maximisation price of the monopolistic producers yields a stable over t, across j and
for all k mark-up: p( j, t ) =
use the monopoly pricing

> . Depending on whether q (1– ) is greater or less than , it will respectively

or the limit pricing p = q  to capture the entire market. In this latter case, the
1
1
leader can capture the entire market by selling at a price slightly below q  (e.g., Grossman and Helpman, 1991,
Ch. 4), since the lowest price that the closest follower can charge without negative profits is .
11
In order to pin down which intermediate goods are produced in each country at t, let: (i)
m(t) and (1– m(t)) be the share of intermediate goods of m-type with production in N and S,
respectively; (ii) m(t) and (1– m(t)) be the share of intermediate goods of m-type with
production in N having overcome imitator and innovator competition, respectively. Hence, the
shares in j [0, 1] production at t with limit price pm,N,N, pm,S,N, and pm,N,S in (9) are
respectively given by m ( 1 – m), m m and 1 – m;10 and a price index for m-type intermediate
goods, pm ( j ) , at t can be defined as a weighted average of the limit prices.
pm ( j )   N  (q  1) m  N  m  m q ( N   S ) .
(10)
Since limit prices in (9) are stable over t, across j and for all k, the problem is
symmetric. This symmetry is dictated by the way in which j enters in final-goods production
and by the fact that all producers of intermediate-goods use the same input. Even without
inter-country protection of patents, the current producer of j enjoys some international
monopoly power, which is temporarily assured by intra-country patent and by costly imitation
in S. The length and magnitude of the monopoly power (measured by mark-ups) are shortened
by international competition, which can be understood as a fall in pricing distortions.
2.4. R&D sector
R&D success in N results in innovative designs to produce intermediate goods, which boost
their quality rung kN = k. R&D success in S means imitation of innovations, since, despite j,
kS  k. As S is not too backward, we assume that there are some top-quality intermediate goods
produced in both countries (that is, for which kS = k). Through trade, S has access to all top
qualities, which improves its probability of successful R&D. However, to capture the world
market in j, S needs to imitate its top quality, which forces firms in S to support the R&D
imitative cost of possibly several quality rungs above their own experience level.
10
The pattern of these shares as functions of the probabilities of R&D has been carried out so that the share
produced in N increases with innovations and falls with imitations (e.g., Dinopoulos and Segerstrom, 2007).
12
Let IN(k, j, t) denote the instantaneous probability of successful innovation in the next
higher quality k ( j, t )  1 of j, and IS(k, j, t) denote the instantaneous probability of successful
imitation of the respective top quality k ( j , t ) ; they are formally represented by
I N (k , j, t )  yN ( j, t )  N q
k N ( j, t )
  N1 q

1
k N ( j, t )
1
 mw,S (t )  mw, N (t )  ;


(11a)
1
1
I S (k , j, t )  y S ( j, t )   S q kS ( j, t )  S1 q k N ( j, t )   mw,S (t )  mw, N (t )   cu (k , j, t ) ;


(11b)
(i) y N ( j, t ) and y S ( j, t ) are the flow of domestic aggregate output resources devoted to R&D
in j at t; thus, this is a lab-equipment model (e.g., Rivera-Batiz and Romer, 1991).
(ii)  N q kN ( j, t ) and  S q kS ( j, t ) , (  N   S  0 ), represent learning-by-past domestic R&D in j at t
(e.g., Grossman and Helpman, 1991, Ch. 12; Connolly, 2003).
(iii)  N1 q kN ( j , t ) ( 1/  ) and  S1 q kN ( j , t ) ( 1/  ) , (  N   S  0 ), are the adverse effects due to the growing
difficulty of new advances at t (e.g., Kortum, 1997).11
1
(iv) mw,S (t )  mw, N (t )  is the adverse effects of market size, which captures the idea that the




difficulty of introducing new quality-adjusted intermediate goods relies on the market size
measured by the employed human capital in N and S because they are sold in both countries.12
(v) cu (k , j, t )  exp ( P  T )  f (k , j, t ) is the catching-up term of S, which sums up positive effects
of imitation capacity and backwardness on IS: exp( P  T ) evaluates the sources of imitation
capacity, which are domestic policies promoting R&D (e.g., Aghion et al., 2001, 2004) and
international trade (e.g., Coe et al., 1997), respectively; function f is formally given by
11
S needs to imitate the top quality and thus  S q kS( j,t )  S1 q  1k N ( j, t )   q k N ( j, t )  1 ( 1) q~( j, t ) to capture the world

S
S
market – where j is an average of all domestic js of its type:
~
QL 
12
QL , S
J
QL , N
J

q
kL,S ( j )
q
kL, N ( j )
~
 q~L , if 0  j  J ; and Q H 
QH , S
1 J
QH , N
1 J

q
kH ,S ( j )
q
kH , N ( j )
 q~H , if J  j  1.
In line with, e.g., Dinopoulos and Segerstrom (1999), this term enables us to rule out scale effects and, due to
human-capital accumulation, is needed for a stable growth rate over t. As it will be made clear below, in (11a, b),
13

0

1
~
f ( k , j , t )  

exp (mw (t ))
~
 ~
2
1  1  exp(m
~ (t ))   Qm ( t )  (1  d ) Qm (t ) 
w
 


d

~
, if 0  Qm(t )  d
~
  2  Qm ( t )
,
~
, if d  Qm(t )  1
(12)
~
Q
~  m  1 and Q
where m
 1 are the S relative level of both employed m-type human
w
m  Q
m
w, S
m,S
w, N
m, N
capital and technological knowledge in m-specific intermediate goods since:
Qm , N (t )   q k N ( j ,t ) 1  dj  Qm (t )   q k ( j ,t ) 1  dj  Qm , S   q k S ( j ,t ) 1  dj ,
J
0
1
J
1
0
J
1
0
(13)
i.e., Qm,N  Qm is the (worldwide) available technological knowledge and Qm,S is the developing
technological knowledge. The first term within square brackets in (12) captures the idea that
employed human capital enhances imitation capacity (e.g., Nelson and Phelps, 1966;
Benhabib and Spiegel, 1994), thereby speeding up convergence with N, since  1  0 affects
how fast the probability of successful imitation rises as the human capital gap falls.
The benefits of backwardness are captured by the 2nd term within square brackets in
(12). The function is quadratic and, once affected by the exponent, yields a rising advantage
of backwardness: the size of  2  Q~m affects the speed at which IS falls as Q~m increases.13
3. Equilibrium
As implied above, R&D firms in N produce innovations, are affected by R&D in S, do not
obtain any royalties for imitation and do not seek to produce abroad. R&D firms in S
undertake imitations, since the imitation cost is less than or equal to that of innovation.
3.1. Equilibrium for given factor levels and level effect of trade
Now, we derive the equilibrium for given aggregate resources allocation, technological
knowledge and human capital. An important feature of the equilibrium is that only one
technology, L or H, is used to produce a particular final good in either S or N.
terms (ii) and (iii) are required for a stable growth rate over t, as in standard models (Barro and Sala-i-Martin,
2004, Ch. 7); term (iv) is also needed for a stable growth rate over t, but due to human-capital accumulation.
13
In line with, e.g., Baumol (1986) and Quah (1997), the rule that the smaller Q~m , the higher the catch up, does
not apply unconditionally: if the gap is not big (if Q~m is above threshold d), S can benefit from a backwardness
advantage (e.g., Barro and Sala-i-Martin, 1997); when the gap is wider
14
~
Qm  d
, backwardness is no an advantage.
When trade starts, an adjustment process in final-goods production occurs. From then
on, each final good is produced by one country in line with the comparative advantage, by one
type of technology. Economic viability of either country and of either type of technology
relies on intra and inter-country productivity and prices of H and L and on intra and intercountry productivity and prices of intermediate goods, due to complementarity in production.
Intra-country prices of H and L rely on the levels supplied for production, Hw(t) and
Lw(t). In relative terms, the obtained intra-country productivity-adjusted quantity of H in
production is
h H w( t )
l Lw( t )
. Inter-country prices of m-type human capital rely on mw,N (t) and mw,S (t),
on AN and AS, and on N and S. In relative terms, the obtained inter-country productivityadjusted quantity of mw,N (t) is given by
    
1
1
mw , N( t )
AN 
N 
AS
S
mw , S( t )
.
Since both countries have access to the top quality intermediate goods, productivity and
prices of these goods also rely on complementarity with either m-type human capital, on the
technological knowledge embodied and on the mark-ups. These determinants are summed up
in the available aggregate quality indexes, QL,N and QH,N . Since in either specific technological
knowledge, the gap is always favourable to N – see (13); even under trade not all innovations
have been already imitated, at each t – the access for S to QL,N and QH,N is a static benefit in
the available technological knowledge and it means that factor-intensity reversal exists.
Endogenous threshold final goods nL , n and nH (implying the use of L-technology in S
for 0  n  nL , L-technology in N for nL  n  n , H-technology in N for n  n  nH , H-technology
in S for nH  n  1 ) follows from equilibrium in the inputs markets. The obtained nL , n and nH ,
relying on determinants of intra and inter-country economic viability of both technologies, are
nL 
z1 z 2 z3
z 2 z3
z 2 (1  z1  z 3 )
, n
and n H 
1  z1 (1  z 2 )  z 2 (1  z 3 )
1  z1 (1  z 2 )  z 2 (1  z3 )
1  z1 (1  z 2 )  z 2 (1  z 3 )

1/ 
  A   Lw, N
where : z1  1   N N 
Lw , S
  AS  S 

1

2 
 
 

1
,
 A  1 /  H w, N
z 2   N N 
H w, S
 AS  S 
15
1
1
2
 QL , N l Lw, N  2 .
 , z3  


 QH , N h H w, N 
(14)
Owing to mw,N and mw,S, each country exports (imports) relatively more final goods that
uses relatively intensively its relatively abundant (scarce) human-capital type (HeckscherOhlian theory). Yet, due to Qm,N, also the Ricardian theory operates. The ratio of index prices
of final goods produced in S and N with H and L-technology are related with nL , n and nH :

 Q (t ) l Lw,S (t ) 
 n (t ) 

   L , N
  L
 Q (t ) h H w (t ) 
p L ,S (t )  1  nH (t ) 
,S
 H ,N

p H ,S (t )

 /2
,
 Q (t ) l Lw, N (t ) 
 n (t )  n L (t ) 

   L, N

 Q (t ) h H w (t ) 
p L , N (t )  n H (t )  n (t ) 
,N
 H ,N

p H , N (t )


p L , S  exp ( ) S (n L )  (n H  n L ) 2

N



 p H , S  exp ( ) S (1  n H )  (n H  n L ) 2


N

 /2
,
;
 p L , N  exp ( ) (n  n L )  (n H  n L ) 2


2
 p H , N  exp ( ) (n H  n ) ( n H  n L )
(15a)
;
(15b)
As will be clear later on, parameters and variables that affect thresholds and index
prices also affect the direction of R&D through the price channel, which, in turn, determines
intra-country wage inequality. It can be shown that, in equilibrium, YS and YN are different due
to employed human-capital levels, exogenous productivity and marginal costs, which directly
affect prices and the comparative advantage in final-goods production.
By the static benefit, S enjoys an abrupt absolute and relative to N benefit in terms of
aggregate output and factor prices. Thus, intra-S and inter-country productivity differences are
instantly affected, but the level effect does not fully equalise inter-country productivity, due
to: international immobility of assets; international differences in exogenous productivity and
in marginal costs; the limited substitutability between types of human-capital types.
The level effect also involves immediate changes in the allocation of resources. In
particular, R&D resources in S increases owing to the increase in: (i) resources; (ii) imitation
incentives, through the positive effect of trade on IS – see (11b)-(v); (iii) markets dimension,
which requires more resources due to the adverse effect of market size on IS – see (11b)-(iv).14
Assuming that, when trade starts, N is relatively H abundant,
HN
LN

HS
LS
and
H w, N
Lw , N

H w, S
Lw , S
,N
produces relatively more H-technology final goods than S. By the operation of the price
14
Resources to R&D immediately increase in N as well, but only for the third reason, (iii). Resources in N are
reallocated at the expense of current consumption, while C in S increases with the level effect.
16
channel there are stronger incentives to improve technological knowledge that saves the
relatively scarce human capital. Despite the drop in the relative price of H-technology final
goods in S when trade starts – see (15a) and considering
QH , N
QL , N

–, it remains greater than in
QH , S
QL , S
N. Due to feedback effects between countries induced by trade, this will stimulate more R&D
directed at improving Northern (worldwide) technological knowledge than in pre-trade North.
Concerning the level effect on wages, the access to better intermediate goods shifts the
demand for m in S upwards; i.e., inter-country wage inequality falls. The absolute (and
relative to N) benefit to L and H is not balanced, though. Complementarity between inputs in
(4) together with the technological-knowledge bias in N,
QH , N
QL , N
, boots the demand for H in S;
thus, the level effect increases intra-S wage inequality (the skilled human-capital premium).
Since trade improves the relative wage of the relatively scarce factor (H, in S), this is against
the standard Stolper-Samuelson effect. Afterwards, the paths of intra and inter-country
productivity differences are affected by the economic structure of both countries since, due to
interaction (feedback) effects, the dynamics (growth effect) involves S and N.
Inter-country wage inequality per unit of m,
of human capital (the relative wage per unit of H),
wm , S
wm , N
, intra-country wage inequality per unit
wH
wL ,
, and intra-country wage inequality per
worker (the relative wage of workers who accumulate H),
wm,S
wm, N
 p A
  m,S S
 pm, N AN
1/ 




1/ 2
, wH   QH , N hLw  ,
wL  QL , N lH w 
Thus, (16) reveals that: (i) wages in N,
wm , S
wm , N
W
wH H w
wL , Lw
, are, respectively:
1/ 2
 Q hH 
W   H ,N w 
 QL ,N lLw 


.
(16)
, are higher, due to the international human-
capital immobility and productivity differences, AN > AS; (ii) relative wages,
wH
wL
and the still
more meaningful W, are driven by the relative demand, which is affected by the Northern
(worldwide) technological-knowledge bias,
employed human-capital structure,
Hw
Lw
QH , N
QL , N
, by the relative productivity,
h
l
, and by the
, which, in turn, is strongly affected by the relative
17
wH
wL
human-capital supply; (iii)
is greater when
QH , N
QL , N
is more skilled biased, employed H is
scarcer and the absolute productivity advantage of H, h, over L, l, is strong;15 (iv) given the
features of the countries,
wH
wL
tends to be greater in S – in line with the empirical literature
(e.g., Beyer et al., 1999; Acemoglu, 2003; Zhu and Trefler, 2005).
3.2. Equilibrium R&D
The expected current value of the flow of profits to the producer of j, V(k, j, t),16 relies on the
profits at t,  (k, j, t), on the equilibrium interest rate and on the expected duration of the flow
(i.e., expected duration of research’s leadership).  (k, j, t) can depend on N = 1, S, pm,N,N ( j ),
pm,S,N ( j ), pm,N,S ( j ) , xn,N (k, j, t) and xn,S (k, j, t) and thus on trade. For example, the expected
duration of the imitator’s leadership counts on IN (k, j, t), which is the potential challenger
(and since the Southern entrant competes with a Northern incumbent), then Vm,S(k, j, t) is:

 s

Vm , S (k , j , t )    m , S (k , j , t ) exp    rs ()  I N (k , j , ) d  ds, m  L for j  [0 , J ], m  H for j  [ J,1] ,


t
 t

(17)
where  (k, j, t), e.g., for a producer of j  [0, J] using an imitation of the top-quality k is:
 L S ( k , j , t )  l  1  


 1
1




1
q k ( j ,t ) (1 )   1  S  DL (t ); DL (t )  Lw, S (t ) AS p L , S (t )   Lw, N (t ) AN p L , N (t ) 






1
.
(18)
Differentiating (17) using Leibniz’s rule, we obtain the dynamic arbitrage equation:
rs (t )  I N (k , j, t ) 
Vm,S (k , j , t )
Vm,S (k , j , t )

 m , S ( k , j, t )
Vm,S (k , j , t )
1  
 k( j, t )
 ln q ,
  
(19)
Plugging (19) in the free-entry R&D equilibrium condition IL,S (k, j, t) VL,S (k, j, t) = yL,S (j, t)
and solving for IN , results, for example, in j  [0, J], the equilibrium probability of successful
innovation, IL,N. Since IN drives the technological-knowledge progress, equilibrium can be
translated into the path of technological knowledge, from which free trade also allows the
15
Our model is thus consistent with the model of Moaz and Moav (2004). In this model, physical-capital
accumulation (in our case technological-knowledge accumulation) has a positive effect on
capital accumulation affects
16
wH
wL
wH
wL
while human-
negatively.
I.e., V(k, j, t) is the market value of the patent or the value of the monopolist firm, owned by consumers.
18
South to benefit. The relationship turns out to yield the well-known expression for the
equilibrium growth rate of QL,N,
Qˆ L, N ,
(considering our example that j  [0, J] and where the
equilibrium L-specific probability of successful R&D, IL,N, given r, pL,S and pL,N is plugged in):

.
1
Qˆ L , N  I L , N q 1    1
(20)
By complementarity between inputs and substitutability between technologies in (4), the
equilibrium growth of available m-type technological-knowledge, translates into the growth
of demand for m-type human capital. Interrelated with the dynamics of international prices of
intermediate goods, p m , and domestic prices of final goods, pm, the growth of wages is:
ˆ m , N (t ) 
w
1

1
  1 ˆ
  1 ˆ
ˆ m , S (t )  pˆ m , S (t )  
pˆ m , N (t )  
 p m (t )  Qˆ m, N (t ) and w
 p m (t )  Qˆ m , N (t ) .




  
(21)
Growth of m-type wages relies on the growth of domestic demand for m-type human
capital that, in turn, relies on the path of the: (i) domestic range of the m-specific technology,
defined by n , nL and nH , which affects prices of (non-tradable) final goods; (ii) world
demand for m-specific intermediate goods, reflected in international prices and driven by
available technological knowledge. From (21) and (3), the domestic equilibrium interest rate
at each t is obtained, which, by Walras’ law, clears the assets market. Then, the general
equilibrium instantaneous growth rate at each t is derived from the Euler equation (2).
3.3. Steady state: growth, pattern of production, prices and wages
The aggregate output has constant returns to scale in available inputs Qm and mw, and Y, C,
1
X  0 xn (k , j, t ) dn
and R  1y( j, t ) dj are all multiples of those inputs. Thus, the stable and unique
0
(common) steady-state endogenous growth rate, which through the Euler equation (2) also
implies a stable steady-state interest rate,
r * ( rL*  rH* ) ,
designed by g * ( g L*  g H* ) is: 17
r*  
*
ˆ*
ˆ* ˆ*
ˆ* ˆ*
ˆ*
ˆ*
ˆ* ˆ*
ˆ*
ˆ*
g *  g *S  g *N  Qˆ m,
S  m w,S  Q m,N  m w, N  YS  Y N  X S  X N  R S  R N  C S  C N 

17
;
(22)
Internally Qˆ L* Lˆ*w = Qˆ H* Hˆ w* and internationally Qˆ m* ,S mˆ w* ,S = Qˆ m* , N mˆ w* , N (due to technological-knowledge diffusion);
moreover, the constant r * is obtained following the same steps as those used to obtain the equilibrium r.
19
g * implies the maintenance of steady-state: (i) technological-knowledge and human-
capital gaps; (ii) final-goods production structure (captured by the thresholds) and prices;
thus, the world productive structure (the world pattern of final goods specialization) tend to be
constant. Indeed, from (22), and considering namely (14) and (15a, b), the result is:
~ˆ
~ˆ
~ˆ
~ˆ
QL*  QH*  L*w  H w*  nˆ *  nˆ L*  nˆ H*  pˆ L* , S  pˆ H* , S  pˆ L* , N  pˆ H* , N  0 ,
(23)
While total convergence in available technological knowledge is abrupt (level effect),
due to trade (of intermediate goods), domestic levels may not converge entirely ( Q~m* and m~ w*
may remain below one): inter-country differences are possible in levels (not in growth rates).
Moreover, in line with Rivera-Batiz and Romer (1991) and Grossman and Helpman (1991),18
trade of final goods (alone) plays no direct role in fostering growth. Indeed, if changes in
prices of final goods from trade of intermediate goods alone to trade of all goods are of 2nd
order19 and there is no increase in R&D,20 there is also no change in the world growth rate.
As the North-South steady state growth rates in wages are equalised – see (21) and (23)
–, a Schumpeterian dynamic equivalent to the static factor-price equalisation (Samuelson)
theorem holds. However, North-South wage levels do not converge totally – i.e., the static
Samuelson theorem does not hold and inter-country wage inequality remains –, due to distinct
non-trade related exogenous productivity and international human-capital immobility.
Inter-country differences and initial conditions H~ w (0)  L~w (0)  1 and
~
~
QH (0)  QL (0)  1 ,
imply:
(i) Qm,S < Qm,N and, since not all innovations have been imitated at t, Q~H*  Q~L*  1; i.e., there is a
drop in the distance to Northern technological knowledge towards the steady state; the drop in
gaps occurs at falling rates since backwardness is less and less advantageous as the steady
18
19
They suggest that free trade of final goods affect the output level but not long-run growth rates.
Evaluation of these changes requires solving numerically the system of differential equations describing
dynamic general transitional dynamics equilibrium by calibration and simulation (see subsection 3.4).
20
We can admit that imitation capacity, evaluated by T in (11b), is enhanced by trade of final goods. If so, the
increase in the steady state world growth rate reflects a kind of scale: the jump in the catching up term due to T.
20
state comes closer;21 (ii) mw,S < mw,N and H~ w*  L~*w  1; i.e., technological-knowledge diffusion
affects human-capital production, thus reducing inter-country gaps; Southern accumulation is
stimulated by higher wages, due to both the fall in the technological-knowledge gap and the
level effect induced by trade, and continuously adjusts the levels to their higher steady-states.
North-South average relative price of final goods produced by H is higher than in pretrade North due to distinct human-capital levels.22 Through the price channel, this induces
higher relative demand for H-specific new designs, biasing R&D in that direction more than
in pre-trade. Such steady state bias – see section 3.4: (i) increases the worldwide supply of Hspecific intermediate goods; (ii) raises n L towards n L* , and falls n and nH until n * and nH* :
i.e., the number of final goods produced with L (H-)technology decreases (increases),23 and S
(N) produces and exports more (less) final goods in steady state;24 (iii) the technologicalknowledge path is vital to the steady state path of relative wages – e.g., (16); it attenuates the
failing intra-N wage inequality path, relative to what would have prevailed under pre-trade,
and, after the level effect, redirects intra-S wage inequality path in favour of falling inequality
– following the slope in N (in line with the data – e.g., Acemoglu, 2003; Robertson, 2004;
Zhu and Trefler, 2005); i.e., trade moderates steady state technological-knowledge bias and
wage inequality paths – due to a Schumpeterian dynamic (since arises from the endogenous
paths of prices that affect R&D) equivalent to the static Stolper-Samuelson result.
3.4. Transitional dynamics, steady-state trade effects and sensitivity analysis
21
The reduction of the gaps reflects different changes in IN and IS. In addition to the advantage-of-backwardness
effect on IS, different changes in IN and IS arise from inter-country differences in the allocation of resources to
R&D. R&D resources increase more in S due to stronger incentives. Incentives remain stronger in the catchingup S as long as the effect of the fall in the imitation cost relative to innovation prevails; i.e., during transition.
22
Indeed, the Southern relative price of final goods produced by H is always higher – see (15a) and (15b).
23
As a result, we can state that a partial Schumpeterian dynamic similar to the static Rybczynski result occurs.
24
Economic development level in S measured by technological-knowledge and human-capital gaps is improved.
21
The system of differential equations describing dynamic equilibrium comes from the
individual utility maximisation with individual optimal time allocation, equilibrium in the
‘labour’ and product markets, and R&D arbitrage conditions (innovation and imitation). By
considering parameter calibration based on empirical literature and on theoretical assumptions
(see Table 1, in Appendix), and by using the 4th order Runge-Kutta method, numerical results
reveal time-path convergence towards the steady state. Figures 2-7, using those results,
illustrate the analysis generated by free trade at t = 0, assuming the starting conditions
and
QH , S
QH , N
LS
HS

LN
HN
 QLL,,NS . They uncover the price-channel effects on: pattern of final goods production;
Q
dynamics of technological knowledge and wage inequality; gaps in technological knowledge
and in human capital. Table 2 (in Appendix) compares initial and steady state values.
The adjustment process is globally stable, and the world steady state is attained at the
end of 145 years. The lengthening in the time needed to attain the steady state indicates that
the transitional dynamic should not be neglected. While worldwide technological knowledge,
QH,N and QL,N, is the same in S and N, South’s domestic technological knowledge, QH,S and QL,S,
remains lower since at each t not all innovations have been imitated. The distance to the
technological-knowledge frontier defines the South’s backwardness. Figures 2a, b show that
~
~
QH and QL increase from initial values (0.30 and 0.35, respectively) towards new steady state
values (0.35 and 0.41, respectively): during the transitional dynamics, technologicalknowledge in S grows more quickly than in N, but slows down until the steady state.
Gaps reduction must thus reflect differentiated changes in the probabilities of successful
innovation and imitation. In addition to the advantage-of-backwardness effect on the
probability of imitation, differentiated changes in the probabilities arise from inter-country
differences in the allocation of R&D resources. Indeed, while increasing in both countries at
rates higher than g * during transition, R&D resources increase more in S due to stronger
22
incentives – reflected in higher interest rates. Incentives remain stronger in the catching-up
South as long as the effect of the fall in the cost of imitation relative to innovation prevails.
~
~
Figures 2a, b also show that H w and Lw increase from the initial values 0.30 and 0.35,
respectively, but at a falling rate until the steady state values 0.46 and 0.54, respectively. The
paths show that, initially and in relation to N, the shares of time dedicated to training in S
increase due to the level effect induced by trade on wages. Thereby, initially the employed
human-capital gaps widen. Only after time t = 7 do the paths change. The path of employed
human-capital gaps is also an indicator of the initially small incentives to do imitation.
Remember that the path of wages depends directly on Northern technological-knowledge
progress and employed human-capital gaps affect the Southern imitation capacity.
~
~ imply that from t = 7 on does S grow at a higher rate
Together, the paths of Qm and m
w
than N, and the differential falls steadily towards the steady state; i.e., S narrows the gaps and
supports higher imitation costs, which pushes the world growth rate down to the steady state.
Figure 2. Transitional dynamics of both human capitals and quality indexes gaps
~
~
~
~
a) Skilled side, Q H and H w
b) Unkilled side, Q L and Lw
0,47
0,55
~
Hw
0,41
~
Lw
~
QH
~
VH (. | w )
0,48
~
QL
0,35
0,41
0,29
0,34
0,23
0
18
36
54
72
90 108 126 144
Time
0,27
0
18
36
54
72
90 108 126 144
Time
Figures 3a, b through 7a, b are arranged in a suitable order to accompany the sequence
of analytical steps that follow. Due to complementarity between inputs and substitutability
between countries and technologies in (4), DN 
23
QH , N H w , N
QL , N Lw , N
and DS 
QH , N H w , S
QL , N Lw , S
are obtained by the
combination of the two types of technological knowledge in N with the respective domestic
human capital. Resulting from the steady-state relationships in (22), such a combination tends
to a constant in each country and, consequently, so do n , n L , nH ,
pH , N
pL , N
and
pH , S
pL , S
– see (23).
The access to state-of-the-art intermediate goods in N, coupled with the scarcity of
skilled human capital in S, implies that DN and DS rise from initial values (1.24 and 1.06,
respectively), but at a decreasing rate, until steady-state values (1.48 and 1.27, respectively) –
see Figures 3a, b. Thus, during transitional dynamics, QH,N H w,Z has outgrown QL,N Lw,Z ; i.e.,
D Z ,
DZ ,
is positive but declines monotonically towards zero. In short, due to trade, DN and DS are
affected by inter-country final-goods price indexes, which, in turn, are affected by employed
human-capital levels (and by ). Moreover, the paths of DN and DS allow us to state that a
temporary partial Schumpeterian dynamic equivalent to the static Rybczynski result holds.
Figure 3. Transitional dynamics of DZ
a) In the innovator country, DN
b) In the imitator country, DS
1,52
1,34
1,44
1,26
1,36
1,18
1,28
1,1
1,02
1,2
0
18
36
54
72
0
90 108 126 144
Time
18
36
54
72
90 108 126 144
Time
The path of each margin between t = 0 (where n  0.454 , nL  0.085 and nH  0.904 ) and
the steady state (where n *  0.433 , nL*  0.096 and n H*  0.882 ) is shown in Figure 4. Since
both countries have access to the same technological knowledge, at each t, each country
produce and export (import) relatively more type of final goods that more intensively use the
relatively abundant (scarce) human capital; i.e., S (N) produce and export relatively more L
24
nL( 0 )
1 nH( 0 )
(H) final goods:
 0.885 
n( 0 ) nL( 0 )
nH( 0 )- n( 0 )
 0.820
and
n L*
1 n H*
 0.814 
n *  n L*
n H* - n *
25
 0.75 ,
where,
consequently, H final goods are relatively more expansive – see Figures 5a, b. As we know
from (16), the paths of thresholds and final-goods price indexes are related – price channel
under international technological-knowledge diffusion. In particular, as a result of the paths of
DN and DS, the path of n indicates that the use of L-technology in the world decreases,
whereas the use of H-technology increases – see Figures 4b, d.26
Figure 4. Transitional dynamics of the thresholds
a) Path of the comparative margin nL
b) Path of the technology margin n
0,099
0,458
0,093
0,451
0,087
0,444
0,081
0,437
0,075
0,43
0
18
36
54
72
0
90 108 126 144
Time
18
1
0,908
0,8
0,901
0,6
0,894
0,4
0,887
0,2
0,88
0
25
Indeed,
nL
and
36
54
n  nL
72
72
90 108 126 144
d) Path of nL , n and n H
0,915
18
54
Time
c) Path of the comparative margin n H
0
36
nH
n
nL
0
90 108 126 144
Time
18
36
54
72
90 108 126 144
Time
are, respectively, the weight of L final goods in S and N, whereas
1  nH
and
nH - n
are,
respectively, the weight of H final goods in S and N; for example, the share of L and H final goods in S increases,
shifting from 8.5% and 9.6% to 9.6% and 11.8%, respectively – see Figures 4a, c, d.
26
This is different from the Heckscher-Ohlin model where ‘factor-intensity reversal’ is absent, but is an expected
result, since inter-country human-capital gaps decrease during the transitional dynamics phase.
25
Despite the level effect in S at t = 0, differences in employed human-capital levels imply
that relative prices of H final goods are always smaller in N:
p*H , N
p*L , N
 0.84 and
p *H , S
p *L , S
p H , N( 0 )
p L , N( 0 )
 0.89 and
p H , S( 0 )
p L , S( 0 )
 0.93 ,
 0.88 – see also Figures 5a, b. Moreover, in line with the paths of DN and
DS, there is a drop in the relative prices of H final goods in both countries. The North-South
average relative price of H final goods (the one that becomes relevant under trade) is higher
than the one prevailing in N alone. Thus, the price channel enhances, from t = 0 on, relative
demand for H-specific new designs, biasing innovation R&D in that direction – see Figure 6.
This bias increases the world supply of H-specific intermediate goods, thus contributing to
increase the number of H final goods and to lowering their relative price in both countries
along the transitional dynamics (e.g., Figures 4b and 5a, b, respectively).
Figure 5. Transitional dynamics of the relative price of H final goods,
a) In the innovator country,
pH , N
pL , N
b) In the imitator country,
0,91
0,95
0,89
0,93
0,87
0,91
0,85
0,89
0,83
0,87
0
18
36
54
72
pH , Z
pL , Z
0
90 108 126 144
Time
18
36
54
72
p H ,S
p L ,S
90 108 126 144
Time
Figure 6 shows the Northern path technological-knowledge bias in autarky and in trade.
Due to complementarity between inputs and substitutability between technologies in (4),
changes in
wH
wL
are closely related to
QH , N
QL , N
– as (16) shows; see also Figures 6 and 7a. The
stimulus to the demand for H, arising from
QH , N
QL , N
induced by trade, increases
wH , N
wL , N
, relative to
what would prevailed under pre-trade. In S, relative scarcity drives a higher pre-trade
which, in turn, increases at t = 0 due to
QH , N
QL , N
wH , S
wL , S
,
– see subsection 3.1. This level effect is reverted
26
in transition to the steady state (see Figure 7a) due to
wH , Z
wL , Z
QH , N
QL , N
in Figure 6. The declining path of
has steady-state dynamics; indeed, once relative prices of final goods attain their stable
steady-state levels, intra-country wage inequality per unit of human capital is governed by the
technological-knowledge bias. In turn, the paths of intra-country wage inequality per worker,
WN and WS, are similar to the DN and DS paths – see (16) – starting from WN 0  1.22 and
WS 0  1.12 , and reaching the constant steady-state values WN*  1.33 and WS*  1.23 .
Figure 6. Transitional dynamics of technological-knowledge bias, ln QQH, N
L, N
0,05
Under trade
-0,03
-0,11
Pre-trade North
-0,19
-0,27
0
18
36
54
72
90
108 126 144
Time
Intra-country wage-inequality paths are compatible with the trends (described by, e.g.,
Richardson, 1995, Caroli and van Reenan, 2001, Aghion, 2002, and Autor et al., 2003) that
point to an increase in wage inequality in favour of H in developed and developing countries.
In our case, such an increase is strongly related to
QH , N
QL , N
, relative to what would prevailed under
pre-trade, which spreads from the developed to the developing country through trade.
For example, due to trade a Schumpeterian dynamic equivalent to the static StolperSamuelson result, the path of
QH , N
QL , N
attenuates the failing path of
wH , N
wL , N
relative to what would
have prevailed under pre-trade, and, only after the positive level effect, redirects the path of
wH , S
wL , S
in favour of declining inequality. International human-capital immobility, and differences
in exogenous productivity levels (directly related with the quality of domestic institutions)
and in human-capital levels between countries imply that there are wage differentials between
27
them, where the largest difference holds in the wages per unit of employed unskilled human
capital:
wH ,S( 0 )
wH ,N( 0 )
 0.71
and
wL ,S( 0 )
wL , N( 0 )
 0.65
,
w*H ,S
w*H , N
 0.57
and
w*L ,S
w*L , N
 0.54
– see also Figure 7b. Moreover, intra-
country wage inequality per worker, WN and WS, increases in both countries.
From the path of the relative (when compared with N) real wages in S – see Figure 7b –,
we can speculate about the welfare change during transitional dynamics. Relative real wages
in S are falling after t = 7 and, thus, the same must hold in Southern welfare.27 It is interesting
to see that, as a result of the increase in the relative wage,
wm , S
wm , N
, until t = 7, final-goods
producers in S lose competitiveness – see Figures 4a, c. The inverse happens after t = 7.
Figure 7. Transitional dynamics of wages
w
b) Differences between countries
a) Skilled premium, ln wH, Z
 
L, Z
0,13
wH,S
wH,N
0,78
With trade in the South
wL,S
wL,N
With trade in the North
0,03
wH,S
p H,N

wH,N
p H,S
0,68
wL,S
p L,N

wL,N
p L,S
-0,07
0,58
-0,17
Pre-trade North
0,48
-0,27
0
18 36
54
0
72 90 108 126 144
18
36 54
72
90 108 126 144
Time
Time
We also checked the robustness of the transitional dynamics results to shocks; i.e., we
analysed via numerical simulations the deviate of one parameter or an initial condition from
its baseline value,28 and we conclude that the model’s qualitative behaviour is similar for the
ranges of values tested. Three comments are worth remarking. Firstly, we confirmed that
changes in DN* and in intra-country wage inequality are related. For example, an increase in h
increases DN* and the skilled wage premium, and also favours economic growth rate and H-
27
Note that as the Southern interest rate must be decreasing during the transitional dynamics, the same is
expected for the movement of real profits from the intermediate-goods sector.
28
Results are available upon request (due to the size of the paper).
28
technology in production (i.e., n * falls). Thus, relative demand for j  [J, 1] rises, enhancing
profits of H-specific R&D and hence biasing technological knowledge in that direction.
Secondly, recalling the human-capital production technology, the growth rate clearly
increases with higher efficiency of either input to the m production, and lower human-capital
depreciation. The more intensive use of the more productive mode (schooling) – either from
increases in intensity H, or enhanced efficiency χF, or higher substitutability  – increases the
relative advantage in accumulating H. The rise in the relative supply of H lowers its relative
wage, but not the relative wage per skilled worker. Price-channel effects explain the changes
in inter-country human-capital gaps. As the relative price of L final goods rises with the more
intensive use of schooling, technological knowledge is directed towards L-specific
intermediate goods. Since S is relatively abundant in L, it benefits relatively more in terms of
productivity, and thus wages, which induces the reduction in human-capital gaps.
Thirdly, in what respects the effect of initial relative levels on steady state, we observe
~
that when S is initially closer to N in one technological-knowledge type – higher QH (0) or
~
Q L (0) – DN* , W N* and WS* fall since imitation is harder with larger technological knowledge in
S, which, in turn, more strongly reduces H-innovative R&D. Besides, relative factor
~
~
abundance implies that more final goods are produced by N (S) when QH (0) ( QL (0) ) is higher.
4. Concluding remarks
By considering trade between the North innovator and the South imitator, this paper connects
the diffusion and the bias of technological-knowledge and, thus, relates the diffusion under
trade with levels, growth rates, specialisation patterns and wage-inequality paths. This
connection is analysed in a model where: (i) mechanisms work through the price of finalgoods channel; (ii) technological-knowledge progress and human-capital production drive
endogenous growth, and their direction, affected by trade, governs wages and specialisation.
29
The comparative advantage is thus induced by a dynamic Ricardian model and by a dynamic
Hechscher-Ohlian model, as it is also reliant on endogenous human-capital.
A process of convergence between countries exists since technological-knowledge and
employed human-capital gaps narrow – i.e., during the transitional dynamics the South grows
at a higher rate than the North, but the differential falls steadily. In spite of the greater
interdependence between countries induced by trade, differences in human-capital levels,
imply that the convergence in prices is only relative: the Southern relative price of skilled
final goods remains higher. Thus, the North-South average relative price of skilled final goods
is higher than in pre-trade North, which increases the relative demand for skilled new designs.
In turn, this boosts the relative worldwide supply of skilled intermediate goods.
Therefore, the interdependence between countries and the price channel, induce more
R&D directed at improving skilled technological knowledge than in pre-trade North, which
originates more moderate paths for technological-knowledge bias and intra-country wage
inequality. In particular, the level effect in the South is reverted and the path of wage
inequality is more moderate than in the pre-trade North. To sum up, a Schumpeterian dynamic
equivalent to the static Stolper-Samuelson result is numerically found. Moreover, as the use
of skilled-technology increases, a temporary partial Schumpeterian dynamic equivalent to the
static Rybczynski theorem occurs; hence, there is ‘factor-intensity reversal’.
Hence, wage-inequality results can be interpreted relative to skill-biased technological
change literature. In that literature, the bias that causes intra-country wage inequality is
mainly induced through the market-size channel. In our case, changes in intra and intercountry wage inequality also arise from the technological-knowledge bias, which is governed
by the price channel under trade and which spreads from the North to the South. In contrast
with the market-size channel, the price-channel process is compatible with the observed
wage-inequality path in favour of skilled labour in developed and developing countries.
30
Due to level and growth effects, trade strongly benefits the South. However, due to the
gaps in exogenous productivity and in human capital, the steady state levels of output and
wages remain different between countries. Indeed, although the level effect strongly favours
Southern wages, South-North wage differences remain: in steady state there is maintenance of
inter-country wage inequality – i.e., they grow at the same rate –, which, in turn, implies that
a dynamic Schumpeterian factor-price equalisation or Samuelson result arises.
In what respects the specialisation patterns, we find that: (i) at each time, both countries
produce, consume and export both types of final goods since they always have both types of
human-capital; (ii) each country exports relatively more final goods of the type that use more
intensively the relatively abundant type of human capital, directly in line with the HechscherOhlian model; (iii) the South (North) produces, consumes and exports imitated (not yet
imitated) intermediate goods, directly in line with the Ricardian model; (iv) the South
produces more final goods at the end of the adjustment process, since it becomes more
developed (the technological knowledge and human capital gaps decrease).
It is also worth recording the fact that numerical simulations show that the model’s
qualitative behaviour is similar if the value of each parameter or initial condition at a time is
changed relative to its baseline value, where slight differences are attained in the value of the
variables in the new steady state. This attests to the stability of the model.
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Appendix
Table 1. Baseline parameter values
Value
AN=N
AS=l=N=S

h
S
T+P
1.60
1.00
0.60
1.20
0.50
0.95
Value
N
S
1
2
4.00
2.00
0.25
0.60
0.10
1.05
d

Value

m
H
F
T

0.03
0.02
0.60
0.09
0.07
0.05
Source: Parameters have been calibrated considering our theoretical assumptions, the literature (e.g., Mansfield et al.,
1981; Mincer and Ofek, 1982; Coe and Helpman, 1995; and Coe et al., 1997; Papageorgiou, 2002; Afonso, 2008) and
the steady-state growth rate of 2.5% (that matches the post-war average growth rate of the U.S. – e.g., Jones, 1995b).
34
Table 2. Comparing initial (initial conditions) and steady state values of the variables
Value at time:
Initial
Steady state
Variable
1.24
1.48
p H,N c)
p L,N
0.89
0.84
DS b)
1.06
1.27
p H,S d)
p L,S
0.93
0.88
0.30
0.35
.049
–.099
0.35
0.41
wH,S
wH,N
0.71
0.57
Hw
~
0.30
0.46
wL,S
wL,N
0.65
0.54
~
0.35
0.54
WN
1.22
1.33
.085
.096
WS
1.12
1.23
.454
.433
Interest rate, r
---
.059
.904
.882
Growth rate, g
---
.028
Lw
nL
n
nH
a)
Value at time:
Initial
Steady state
DN a)
~
QH
~
QL
Notes:
Variable
QH , N ( t 0 )
QL , N ( t 0 )
p*L,N  0.790 ;
d)

QH , N ( 0 )
QL , N ( 0 )
 1.05 ;
b)
QH , S ( 0 )
QL , S ( 0 )
 0.90
ln
c)
QH,N
QL,N
*
and
pH,N (0)  0.698, pL, N (0)  0.786, p H,
N  0.665
pH,S (0)  0.885, pL,S (0)  0.951 , p *H,S  0.741 and p*L,S  0.841 .
35