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ECON/SØK 4415 International trade
Lecture 12: Technology, trade and growth
Technology and trade
Readings: Lucas (1988), pp. 27- 35 and Rodriguez and Rodrik (1999).
Trade and growth in the Ricardian model
Much new growth theory is based on either R&D or external ‘learning by
doing’. These models have in common that they generate constant growth
rates as a result of economic mechanisms. This contrasts exogenous
growth models that assume growth in the long run to be generated by
exogenous forces.
Neo-classical versus endogenous growth
Y-production, K- capital, L- labour,
s- a constant savingsrate,
a ’dot’ above a variable denotes derivative with respect to time, so
K  sY
Neo-classical growth
Y  AK  L1
  K  1 L 
 sY L 
Y L A
K
L
        
     sA   ,
 L
Y L A
K
L
L 
 K L

Growth will decrease with the amount of capital
iff
A
0
A
Endogenous growth

Yi  AK i Li
1
Y   AK i Li

, K   Ki ,
1

K
A  A  ,     1
L
 A K    L1    A K
Y L K L  sY L  
L 
    
    sA  
Y L K L  K L 
L
Growth is independent of the amount of capital. How could this be the
case?
In endogenous growth theory, very often some kind of externality is
assumed, so that investments of one firm increase the productivity of
other firms.
Learning by doing
Lucas (1988) incorporates these features of technology into a model with
two sector and international trade.
Assume a two-goods (c1 and c2) economy. Let production be given as
ci=hiuiN
ci denotes production of good i. hi denotes the human capital in this
industry. ui denotes the share of the total labour force in the economy
used in this industry. N is the total labour force. Therefore, uiN is the use
of labour in industry i. It is seen that there are constant returns to scale in
this industry since use of labour per unit is given by uiN/ci=1/hi which
will be treated as a constant for each firm.
Also note that u1+u2=1.
Now assume that hi increases as a function of the effort ui devoted to
producing good i. We write
hi  hi  i ui
The ‘dot’ above a variable denotes this variable derivated with respect to
time. Therefore growth rates are given by:
hi
  i ui
hi
It is clear from the above that growth rates in both industries will be
proportional to production.
It is important in the present model, that learning effects from production
is assumed to be exogenous to each firm. Human capital available for
each industry increases by the size of this industry (measured by hi), but
these effects may be too small for individual firms to take them into
account.
Now assume that one industry, good 1, is a high-technology industry.
That is δ1>δ2. This implies that the potential for growth in this industry
will be higher than in industry 2.
Now, the intuition in the model to presented is the following: The
differential equations for productivity growth in each industry generate
endogenous economic growth. If the economy is such that the two goods
are close substitutes, in autarky the industry with the highest growth, will
take over the economy. If the two goods are poor substitutes, growth in
the high technology industry will free resources to be used in the low
growth industry. In a border case, the two industries will maintain their
share of the economy.
Now if two countries with the above technologies start to trade, it might
be the case that they specialise according to short run comparative
advantages. However, one of the industries will have higher growth
potential than the other. In some cases this will cause a country
specialised in a low-tech industry to loose from trade.
Autarky
Now, we first take a look at the economy in autarky:
Assume that consumers have the CES-utility function

U (c1 , c2 )   1c1

  2 c2 

1 / 
Above, αi>0, α1+α2=1, ρ>-1 and σ=1/(1+ρ) is the elasticity of
substitution. This implies that 0<σ<∞. Note that this equation is the same
type of CES utility function used in many of the lectures. The relationship
between the CES utility function and the Cobb-Douglas function is
discussed in the appendix.
Let the first good serve as numeraire and the let (1, q) be prices. In this
case we get from utility maximisation that:

L   1c1

  2 c 2 

1 / 
  c1  c 2 q  Y 
1 

L
  1c1    2 c 2     1c1  1    0
c1
1 

L
  1c1    2 c 2     2 c 2  1  q  0
c 2

 1c1  1 1

 2 c 2  1 q
c1  1  2 


c 2  q  1 
1

1
1 

1
c 2   2  1  1    2  
  q
   q
c1   1 
 1 
Above Y denotes income. The relative demand functions imply that both
goods will be produced.
Since there is perfect competition in this economy, relative prices will
equal relative costs. Use of labour per produced unit equals 1/hi.
Therefore relative costs of good one to good two is h1/h2. We get q=h1/h2.
From this we get:
c 2 u 2 h2   2 

 
c1 u1 h1   1 

 h2 
 
 h1 

The first equation is given from the supply-side while the second
equation is given from the demand side. Using the fact that u2=1-u1, we
get:
1  u1   2 
  
u1
 1 

 h2 
 
 h1 
 1
Therefore:
u1 
1
 
1   2 
 1 

 h2 
 
 h1 

 1
 1
  2   h2 
   

h
u 2   1   1   1
   h 
1   2   2 
  1   h1 
We have relative prices as q=h1/h2. From this we get the growth rate for
prices:
q h1 h2  h1 h2 h2 h1 h2

 
  1u1   2 (1  u1 )   1   2 u1   2
q
h1 h1 h2
h22
1
   

  1   2 1   2  q1    2
   1 

Note that this equation is a first order linear differential equation. Given
initial endowments of h1(0) and h2(0), this equation determines the
allocation of the workforce at each date and therefore, the paths of h1(t)
and h2(t).
q is stationary when
   

1   2  q1 
   1 

1

2
 1   2 
Furthermore, this boundary is between zero and unity. When q is so that
the left hand side is larger than the boundary, there is positive growth.
When q is so that the left hand side is smaller than the boundary, there is
negative growth.
Now differentiate the boundary w.r.t q:

d    2  1 
1    q 
dq    1 


1   q
1


2 
 
 1 

   

1   2  q 1 
   1 

2
From this it is seen that when σ<1 the boundary is decreasing in q and
when σ>1 it is increasing in q.
From this we can illustrate the dynamics of q in the following diagram:
1
1+(α2/α1)σq1-σ]-1
(σ>1)
δ2/(δ1+δ2)
1+(α2/α1)σq1-σ]-1
(σ<1)
q*
In the case when σ>1, the slope is positive. This is the case when the two
goods are substitutes. That is, when the price of one good increases,
consumption of the other good eventually increases more than
proportionally. In this case, to the left of q*, dq/dt <0 and q tends to zero.
To the right of q*, dq/dt>0 and q grows without bounds. Therefore the
system is unstable and in autarky the economy converges to complete
specialisation in one of the two goods. The choice between the two goods
depends on the initial situation. If the economy is initially good at
producing good 1, it produces a lot of it. After a while the economy
becomes specialised in this good and produces smaller and smaller
amounts of good 2.
If the goods are poor substitutes, σ<1, the slope is negative. This is the
situation indicated by the dashed line. In this situation q* is stable and the
workforce will be allocated so that δ1u1=δ2u2.
If σ=1 the curve is flat. The workforce will be allocated as dictated by the
demand weights, ui=αi, and this allocation is maintained forever. The
autarky price growth or (shrinks) at the constant rate (dq/dt)/dq=α1δ1-α2δ2
forever.
Lucas describes the situation for different values of σ by the following:
‘As we learn how to produce computers more and more cheaply, then, we
can substitute in their favor and consume more calculations and fewer
potatoes (σ>1), or we can use this benefit to release resources from
computer production so as to consume more potatoes as well (σ<1)’.
It is important here that the equilibrium paths may not be efficient.
Agents do not take into account learning effects. If they did, they would
allocate more resources into the high δ good.
Trade
Now introduce trade into this model. Assume that there is perfectly free
trade in the two final goods and a continuum of small countries. In that
case the world prices are (1,p) and p is taken as given for each firm and
each country.
Now, when there is free trade, comparative advantages determine
specialisation (as in the static case). Therefore, some countries become
specialised in the high tech good (c1) and some countries become
specialised in the low tech good (c2). In the dynamic case however,
comparative advantages become strengthened over time as a result of
learning by doing. In the figure below, countries which have endowments
above the price line specialise in production of good 2 while countries
which have endowments below the price line specialise in good 1.
In the countries that become specialised in good 1, h1 grows and h2 is
constant. In the countries that become specialised in good 2, h2 grows and
h1 is constant. Therefore endowments develop as indicated by the arrows.
h2
1
p
h1
As endowments change, so do the prices of the goods. Could these price
changes induce any country to change its specialisation? It is clear that if
any country changes, it must be a country producing good 1 that switches
to produce good 2. The reason is that there is higher growth in the good 1
sector leading to lower prices for this good. Terms of trade are moving
against good 1.
From the demand side we have:

c 2   2  
  p
c1   1 

c 
p   2 
 c1 


1

2 
 
 1 
p
1 c 
   2 
p
  c1 

1
 1

 c2 c1  1
     1   2 
 c 2 c1  
If productivity increase in good one is lower than this, so that
1 
1

 1   2 

  1
,
2
1
producers of the high tech good will shift. From now on, assume that the
opposite is the case. That is, σ≥1-δ2/δ1. In this case no country shifts and
the dynamics of prices can be read off the relative demand schedule:
p
1 c 
   2 
p
  c1 
1
 1

 c2 c1  1
     1   2 
 c2 c1  
Now we are in the position to deduce the growth rates of the two types of
countries. Countries specialised in good one will grow at the rate δ1.
Output of good 2 producers, in terms of good 1, growth at the rate
2 
 2
p
 2  1
p

Generally therefore, countries will have constant, but not equal growth
rates.
The condition that high tech producers grow fastest is that
1   2 
1   2

This is equivalent to the condition that σ>1. That is, producing high-tech
goods will lead to higher than average growth rates if the two goods are
substitutes.
Therefore it matters in which industry a country becomes specialised. If
you specialise in the ‘wrong’ industry, you may experience lower growth
than if you specialise in the ‘right’ industry. Note, however, that this
effect will, at least partially, be offset by price decreases in the industry
with the highest growth rate.
In any case, learning by doing effects may constitute a case for trade
policy. A country might increase its long term growth rate if it manages
to specialise in the high-tech industry.
A model of learning by doing and trade policy
An application of theories on learning by doing is taken from Rodrik and
Rodriguez (1999). The model is for a small open economy. It has two
sectors, manufacturing (m) and agriculture (a). Labour is the only factor
of production and the size of the labour force is set equal to one. n
denotes the share of the labour force used in manufacturing. Production is
characterised by decreasing returns to scale at the firm level. Production
in the two sectors are given by:
1)
X m  An
2)
X a  (1  n)
A denotes total factor productivity in manufacturing. It evolves according
to
1)
A  An

3)
A
 n 
A
The authorities in this country can choose to impose a tariff, ‘tau’, on
manufacturing production. This implies that prices on manufacturing
goods (which they import) can be raised above the prices without the
tariff. Manufacturing will therefore be able to increase production.
An equilibrium condition in the economy is that the marginal productivity
of labour net of the tariffs equals between the sectors. We therefore have:
4)
1  n  1  1   An 1
Now differentiate eq. 4. w.r.t. n and ‘tau’:
   11  n 
 2
dn  An 1 d  (1   ) A(  1)n  2 dn

dn
An 1

0
d 1   1  n   2  1   A1   n 2
We therefore obtain the result that the share of the labour force working
in industry will increase if one increases the tariff rate.
Derivate eq. 4. w.r.t. time to obtain:
   11  n 
 2
n  1   A n 1    11   An 2 n
Insert for ‘A dot’ to obtain:
   11  n 
 2
n  1   An n 1    11   An  2 n
and insert from eq. 4 to obtain:
   11  n 
 2
 1
 1
n  n  1  n     11  n 
n
n
Solve for the growth rate of n to obtain:
n n 1  n 

n
1
5)
At world market prices, income level is
Y  An  n  1

6)
Now derivate this w.r.t. time to obtain:
 1
Y  A n  An 1 n   n  1 n

Y
 1
A n An 1 n  n  1 n



Y
Y
Y
Y
Define the share of manufacturing in GDP as
An
,
Y


1  n
1  
Y

From this we obtain
Y A
n n(1   ) n
    

Y A
n n(1  n) n
n    

n    

So
n 
n1   
 1  n n





n



1  n  
1   
n 

 1  n   n1   
1   


n1   
  1  n  


Y


  n
 n  
Y
1


7)
Now we have:
1  n  1  1   An 1
Y
Y

1 

 1   
1 n
n

n   1     n1     n
Now, if ‘tau’=0, we have that n=. Therefore
Y
 n  n
Y
Since n increases with ‘tau’, growth will increase with ‘tau’ when ‘tau’ is
zero. From eq. 7
7)
Y


  n
 n  
Y
1


when <n, the second term becomes negative. Therefore, the positive
growth effects of tariffs taper off when the size of the tariffs increases.
The intuitive reason is the following:
A positive tariff draws resources into the manufacturing sector and
enlarges the scope of ‘learning effects’. A tariff also imposes a static
efficiency loss. The cost of this loss is larger the larger is the
manufacturing sector.
Appendix: Cobb-Douglas as special case of CES
A CES-function in two goods, x1 og x2, be written

U  x1  (1   ) x 2

1

In the text ρ is the negative of ρ here. The elastitcity of substistuion is  =
1/(1-) and very often  = (-1)/ is used. α denotes expenditure shares
for each good. For many CES-functions these are assumed equal for each
good and therefore disregarded. The function above is more general in the
sense that each good is allowed to have different expenditure shares.
If  approaches one,  approaches zero. In this case we get a function of
the type [1]. This is not a defined expression. We therefore reformulate
the fuction to one of the type (0/0) which can be analyzed with use of
l’Hopital’s rule:
lim ln U  lim
 0
 0
1



ln x1  (1   ) x 2 
x1 ln x1  1   x 2 ln x 2
  ln x1  1    ln x 2
 0
x1  1   x 2
lim

U  e ln1  (1 ) ln x2  x1 x12
In the second equation I derivated w.r.t ρ in the numerator and the
denominator. In the next I let ρ approach zero. The last equation shows
the result.