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Supplementary Notes Lecture 1. Ricardian Models
Address the following issues:
 Why do wages differ so much across the world?
 Does economic growth in the “Third World” hurts the “First World” prosperity?
 Should developed countries stop trading with developing countries?
Explain the basis for trade:
 Countries trade with each other because they are different from each other.
 Comparative advantage is an essential concept.
 In Ricardian models comparative advantage is solely due to international differences in the
productivity of labor.
David Ricardo
Figure 3.1 David Ricardo (1772-1823)
Born in London as the third son of a Jewish family emigrated from Holland he married
the daughter of a Quaker and was disinherited by his parents. Ricardo nonetheless
accumulated a fortune as a stock-jobber and loan contractor. As Blaug (1986, p. 201) puts
it: "Ricardo may or may not be the greatest economist that ever lived, but he was certainly the
richest." His fame today rests mainly, of course, on his contributions to the theory of
comparative advantage.
David Ricardo and his principle of comparative advantage
Like in the case of Smith’s principle of the absolute advantage Ricardo’s concept of comparative advantage
applies both to nations and individuals. What if one party does not have an absolute advantage in anything?
Can people still cooperate with each other and benefit from their cooperation?
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Numerical Example. Trade between England and Portugal in cloth and wine
England
Portugal
Total
Labor hours necessary to produce 1 unit of Cloth 1
2
(i.e. industrial good)
Labor hours necessary to produce 1 unit of Wine 2
8
(i.e. agricultural good)
Potential output of cloth given total labor
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12
12
endowment – 24 hours
Potential output of wine given total labor
12
4
12
endowment – 24 hours
Actual preferences concerning allocation of time 8
4
12
to the production of cloth - 8 hours
Actual preferences concerning allocation of time 8
2
10
to the production of wine - 16 hours
Trade in cloth
8
8
No gains
(imported) (exported)
Trade in wine
3
3
England gains 1,
(exported) (imported) Portugal gains 1
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Simple Ricardian 2x2x1 model
Basic Assumptions:
Two countries: Home and Foreign (*)
Two goods (homogenous): 1 - agricultural (wine) and 2- industrial (cloth)
One factor of production: Labor
Technology: linearly homogenous production function (CRS), fixed labor input coefficients in both sectors
Production function in sector i can be written as:
Qi 
Li
ai
where:
Qi – output
Li – labor input
ai – unit labor requirement (i.e. the number of hours you need to work to obtain 1 unit of good i)
This production function implies that marginal and average products of labor are constant and equal:
1
APL  MPL 
ai
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Labor supplies in both countries L and L* are inelastic (do not change with changing wage rates)
Perfect competition prevails in the economy in product and factor markets (no markup over the marginal
cost and wages equal to the marginal product of labor) in both countries
People want to consume both goods (decreasing marginal utility of consumption of each good)
All people are homogenous (only one group, same tastes, equally productive, supply the same amount of
labor)
Autarky Equilibrium (No trade)
SUPPLY SIDE:
Profit maximizing firms in sector i take as given both product prices (pi) and the wage rate (wi)
Profit = Sales revenue – labor costs
p

L
i  pi Qi  wi Li  pi i  wi Li   i  wi  Li
ai
 ai

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Demand for labor in sector i can be obtained from the First Order Condition (F.O.C.)
 i
p
 i  wi  0
Li
ai
pi
 wi
ai
(interpretation: LHS = the value of the marginal product of labor, RHS = nominal wage)
Alternatively,
1 wi

ai pi
(interpretation: LHS = marginal product of labor, RHS = real wage in terms of i)
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pi
 wi
ai
p
Demand for labor Li = (0, )if i  wi
ai
p
if i  wi
ai
p
0if i  wi
ai
p
Output Qi = (0, )if i  wi
ai
p
if i  wi
ai
0if
Remember that the supply of labor is fixed, hence this limits the amount of output produced.
Full employment condition:
L1  L2  L
Alternatively, (using the production functions)
a1Q1  a2Q2  L (PPF)
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Production possibility frontier (PPF) is derived from the full employment condition
Workers are mobile between sectors which implies the equalization of the nominal wage rate:
w1 = w2 = w
If both goods are to be produced in the economy then the values of marginal products of labor in both sectors
must be equalized:
p1
p
w 2
a1
a2
Case 1. If
p1 p2
p
L

 w  2 , L1  0, L2  L , Q1  0, Q2 
a1 a2
a2
a2
Case 2. If
p1 p2
p
p
L
L
L  L1
(positive employment in both sectors)

 w  2  1 , Q1  1 , Q2  2 
a1 a2
a2 a1
a1
a2
a2
Case 3. If
p1 p2
p
L

 w  1 , L1  L , L2  0, Q1  , Q2  0
a1 a2
a1
a1
(everybody employed in sector 2)
(everybody employed in sector 1)
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The EQUILIBRIUM wage rate is equal to the maximum value of the marginal product of labor
(if both goods are produced then the values of marginal products will be equalized).
Assuming that both goods are produced, if you want to increase the output of one good you will have to
decrease the output of the other good. This can be illustrated using the TRANSFORMATION CURVE
called also the PRODUCTION POSSIBILITY FRONTIER (PPF). This curve shows you the maximum
output of one good, given the output of the other good.
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Q1
L
a1
L
a2
Q2
Figure 1. Production Possibility Frontier
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You can differentiate totally the full employment condition to obtain the slope of the PPF (transformation
curve):
dL  a1dQ1  a2 dQ2  da1Q1  da2Q2
Knowing that dL  0 (total labor supply remains unchanged) and da1  da2  0 (technology remains
unchanged) we get:
dQ1
a
 2
dQ2
a1
dQ1 a2
(marginal rate of product transformation – MRPT) is the opportunity cost of

dQ2 a1
good 2 expressed in terms of good 1.
Recall that MRT  
The opportunity cost of good 2 is equal to the amount of output of good 1 our economy would have to give
up (to release resources needed) to produce an additional unit of good 2.
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Example. England and Portugal
In our example in order to produce one unit of good 2 in Home country (England) you need 1 hour of work
a2 = 1 and in order to produce one unit of good 1 you need two hours of work a1 = 2. In this case the
opportunity cost of good 2 expressed in terms of good 1 equals ½. (i.e. in order to produce an additional unit
of good 2 in England you need to give up ½ of a unit of good 1.) In Foreign country (Portugal) you need 2
hours of work to produce one unit of good 2 and 8 hours of work to produce one unit of good 1. Hence the
opportunity cost of producing good 2 in terms of good 1 equals 1/4. Comparing opportunity costs in both
countries we can notice that the opportunity cost of producing good 2 (cloth) expressed in terms of good 1
(wine) is lower in Portugal than in England.
In our example, when the production possibility frontier (transformation curve) is linear, the opportunity cost
is constant. (However, it will not be constant in other, more complex models that we will discuss later in
class).
DEMAND SIDE:
GDP = Consumer Expenditure = Labor Income (No profit by assumption – perfect competition)
Consumer budget constraint:
p1C1  p2C2  wL  p1Q1  p2Q2
Consumer expenditure = labor income = sales revenue (firm income)
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The slope of the budget constraint can be obtained by total differentiation:
p1dC1  p2 dC2  dp1C1  dp2C2  dwL  wdL
Knowing that dL  0 (total labor supply remains unchanged), dw = 0 (the wage rate remains unchanged) and
dp1  dp2  0 (product prices remain unchanged) we get:
dC1
P
 2
dC2
P1
Note that if both goods are produced the slope of the budget constraint equals the slope of the transformation
curve. Why? Because marginal products of labor will have to be equalized if labor moves between sectors!
p1 p2

a1 a2
Hence,
dC1
P
a
dQ
 2  2  1
dC2
P1
a1 dQ2
However, if the marginal product of labor in sector 1 is smaller than the marginal product of labor in sector 2
then the slope of the budget constraint is bigger and only good 2 is produced
 p1   p2   p2   a2 
          
 a1   a2   p1   a1 
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L
a1
L
a2
Figure 2. Production possibility frontier and budget constraint (only good 2 is produced)
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In order to assure that both goods are produced in the closed economy Inada conditions must be satisfied:
U1 (0, C2 ) 
U (0, C2 )

C1
U 2 (C1 ,0) 
U (C1 ,0)

C2
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L
a1
C
C1
C2
Figure 3. Equilibrium in the closed economy
L
a2
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Summary of conclusions (closed economy):
In autarky the price ratio is determined by the supply side only (unit labor requirements) and consumer
preferences do not matter for relative price determination. However, consumer preferences do matter for
allocation of labor across sectors (and the amounts of each good produced).
p1 p2
p
a

 w  pA  2  2
a1 a2
p1 a1
(the slope of PPF)
OPEN ECONOMY
Now imagine that there is FOREIGN country where the autarky price ratio (relative price) is lower than in
HOME country.
pA 
p2 a2 a2 * p2 *
 

 pA *
p1 a1 a1 * p1 *
This implies that Home country has comparative advantage in production of good 1, and Foreign country has
comparative advantage in production of good 2.
How do we determine relative prices (p2/p1) in a trading equilibrium?
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TOTAL WORLD SUPPLY = TOTAL WORLD DEMAND
Q1W  Q1  Q1*  C1  C1*  C1W
Q2W  Q2  Q2 *  C2  C2 *  C2W
Constructing the RELATIVE SUPPLY CURVE
CASE 1. Both countries are small
If the relative price of good 2 is below the opportunity cost in FOREIGN country (and also in HOME
country), no country produces good 2, both countries produce good 1.
p2 a2 * a2


p1 a1 * a1
 a *
Relative price range  0, 2 
 a1 * 
L L*
Output Q2W  0, andQ1W  Q1  Q1*  
a1 a1 *
Q2W
Relative output W  0
Q1
CASE 2. Home country (Country 1) is small, Foreign country (Country 2) is large
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p2 a2 * a2


p1 a1 * a1
Relative price equal to the autarky price in Foreign country (2), hence country 2 produces both goods
(incomplete specialization in production), while Home country produces only good 1
Relative output range (0,
L * / a2 *
)
L / a1
CASE 3. Both countries are large (complete specialization in production)
a2 * p2 a2


a1 * p1 a1
Foreign country produces good 2, Home country produces good 1.
a * a 
Relative price range  2 , 2 
 a1 * a1 
L * / a2 *
Relative output
L / a1
CASE 4. Home country is large, Foreign country is small
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a2 * p2 a2


a1 * p1 a1
Relative price equals the autarky price in Home country, Home country produces both goods (incomplete
specialization in production), while Foreign country produces only good 2.
Relative output range (
L * / a2 *
, )
L / a1
The RELATIVE DEMAND CURVE
Traditional, negatively sloped, the negative slope reflects the substitution effect, if the relative price of good
2 increases, its relative demand falls, the location of the relative demand curve depends on the relative
country size (which determines relative demand)
The equilibrium relative price is determined by the intersection of the relative demand and the relative
supply curves.
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p2
p1
a2
a1
a2 *
a1 *
L * / a2 *
L / a1
Q2W
Q1W
Figure 4. Relative demand and relative supply
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GAINS FROM TRADE
Now we demonstrate that countries can gain from engaging in international trade.
Assume the complete specialization case.
When Home country (country 1) specializes in production of good 1 the world price of good 1 equals
p1  a1w
When Foreign country (country 2) specializes in production of good 2 the world price of good 2 equals
p2  a2 * w *
Hence, the relative price under complete specialization equals
p2 a2 * w *

p1
a1 w
The ratio of unit labor requirements is fixed, hence it is the relative wage that adjusts to equilibrate the world
product markets!
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Now look at behavior of real wages (expressed in terms of both goods = their purchasing power)
Real wage in Home country expressed in terms of good 1 after opening to trade remains unchanged (as it is
determined only by the Home country technology level)
w 1

p1 a1
However, real wage expressed in terms of good 2 will change!
w w p1 1 p1


p2 p1 p2 a1 p2
After opening to trade the relative price of good 2 (expressed in terms of good 1) in Home country is lower
(compared to autarky). So our wage rate expressed in terms of good 2 is higher as we can consume more.
p A  pT  p A *
Similar results hold for Foreign country (i.e. real wage expressed in terms of good 2 remains unchanged
while increases in terms of good 1).
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L
a1
A
C1T
C
C1
B
C2
C2T
L
a2
Figure 5. Graphical illustration of gains from trade
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Decomposition of the gains from trade:
 Gains from exchange (move from C to B) that come from the change in relative prices
 Gains from specialization (move from B to A) that come from adjusting the levels of output
Extensions of the Ricardian model
Actual trade is characterized by the exchange of a large number of goods (more than 2!). For example, the
SITC nomenclature has 3118 5-digit product categories. However, the number of goods actually traded is
larger. Some commodities like textiles are well represented (200 entries) but trade classification does not
fully account for the degree of product differentiation of many other items (such as bolt, cars, etc.).
Many goods DFS77 model
Now we want to extend the Ricardian model to the case of many goods using the continuum assumption
originally developed in the paper by Dornbusch, Fisher and Samuelson (1977), published in the American
Economic Review. To simplify the analysis the model assumes a continuous rather than discrete number of
goods. The model allows to study the effects of growth, demand shifts and exogenous technological change.
In each case the focus of the analysis is to determine:
i)
ii)
the dividing line between exported and imported goods,
the position of the relative wage that assures balanced trade.
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SUPPLY SIDE
The model assumes that each good is produced with constant unit labor requirement both at home and
abroad. For i-th good, ai represents the unit labor requirement in the home country, while a i* the unit labor
requirement in the foreign country.
Goods can be ranked according to the diminishing home country comparative advantage. Hence, if n goods
are produced we have:
a1 *
a1

strongest
comparative
advantage

a *
a *
a2 *
 ...  i  ... n
a2
ai
an

weakest
comparative
advantage
In working with a continuum of goods we index commodities on an interval, say [0,1], in accordance with
diminishing home country comparative advantage. A commodity z is associated with each point on the
interval, and for each commodity there are unit labor requirements in both countries: a(z) and a*(z). Define
A(z) = a*(z)/a(z) as the ratio of foreign to domestic unit labor requirement for commodity z. Assume that
A(z) is a smooth, continuous function decreasing in z, i.e. A’(z) < 0. The function A(z) can be graphed in
Figure 1 as a downward sloping schedule against z varying between 0 and 1.
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A(z)
0
1
z
Figure 1. Relative unit requirement function
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Now consider the range of goods produced in the home country (and those produced abroad).
It will be cheaper to produce the good in the home country if unit production cost is smaller than abroad:
a(z)w ≤ a*(z)w*
This is called the efficient specialization condition. If we define relative wage as w/w* = ω, this condition
can be rewritten as:
ω ≤ A(z)
In other words, the good will be produced in the home country if the relative wage is smaller or equal the
relative productivity.
For a given relative wage the home country will produce the range of commodities:
0 z ~
z ()
while the foreign country will produce the range of commodities:
~
z ()  z  1
If we know the relative wage ω we can determine the borderline commodity ~
z.
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A(z)

0
z
1
z
Figure 2. The pattern of international specialization given the state of technology and relative wages.
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DEMAND SIDE
On the demand side we assume homothetic and identical consumer preferences in both countries.
In particular, we assume that the demand functions are derived from a Cobb-Douglas utility function.
In the discrete case we would have:

bi

constant
expenditure
share
pi ci
,
Y
n
where:  bi  1, and bi = bi*
i 1
In the continuous case we have:
b( z ) 
p ( z )c ( z )
,
Y
1
where:  b( z )dz  1, and b(z) = b*(z).
0
and where:
Y – total income
c(z) – consumption demand for good z
p(z) – price of good z
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Now, define the fraction of income spent (in both countries) on those goods in which the home country has a
comparative advantage.
~
z
v( ~
z )   b( z )dz  0 ,
where v' (~
z )  b(~
z )  0 , and 0  v(~
z )  1.
0
In the similar manner the fraction of income spent on foreign goods can be defined as:
1
1  v(~
z )   b( z )dz
~
z
EQUILIBRIUM
We can use the market equilibrium condition to determine relative wage and international specialization in
production. Equilibrium in the market for goods produced in the home country requires that the total value of
spending on home-made goods equals the domestic labor income (due to the fact that all sales revenue is
paid out to the workers, there is no profit by assumption – perfect competition). Hence:
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v
(~
z ) [ wL  w * L*]  wL


 
 sales
fraction
of
income
spent
on
home  made
goods
total
world
income
revenue
( our
income)

total
world
expenditure
on
our
goods
The above condition associates with each z an value of the relative wage w/w*. The alternative interpretation
of this condition is as follows:
[1  v( ~
z )] wL


 our 
share
income
of
(exp enditure)
imports
in
our
exp enditure



IMPORTS
 v
(~
z) w
* L*

foreign
share
income
of
home  made expenditur e
goods
in
foreign
expenditur e

EXPORTS
Hence, the above condition can be interpreted as the balanced trade condition.
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This condition can be rewritten as:
w  v( ~
z)  L*
L*
~



B
(
z
,
)
w * 1  v( ~
z )  L
L
B schedule is upward sloping because an increase in the range of commodities produced at home (at constant
relative wages) lowers our imports and raises our exports. The resulting trade imbalance has to be eliminated
by an increase in our relative wage (that would raise our imports and reduce exports).
The alternative interpretation (home labor market interpretation).
If the number of goods produced at home increases (but wages remain unchanged) then the demand for
home labor would increase (with increasing number of goods produced at home) and the demand for foreign
labor would decrease. In this case the relative wage has to increase to equate the demand for domestic labor
to the existing (fixed) labor supply.
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B(z)
DEFICIT
'

SURPLUS
0
z
z'
1
z
Figure 3. The balanced trade condition.
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A(z),
B(Z)

0
z
1
z
The final step is to combine the efficient specialization condition and the balanced trade condition.
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