Download Equation Chapter 1 Section 1A detailed explanation of Kruman

A detailed explanation of Kruman (1980)
trade model with impeded trade.
Notation
ci - consumption of variety i
xi - production of variety i
 - fixed labor requirement
 - variable labor requirement
L, L - labor endowment of the home and foreign country
g - the portion of the shipped good that arrives to destination
c - consumption of the foreign variety in the foreign country
ĉ - consumption of the foreign variety in the home country
x - output of the foreign variety
 - preference parameter such that  
1
is the elasticity of substitution
1
p - factory price of the domestic varieties in the home country
p̂ - delivered price of the domestic varieties in the foreign country
p  - factory price of the foreign varieties in the foreign country
p̂  - delivered price of the foreign varieties in the home country
1
Setup
Consumers maximize
U   ci
i
Notice that in this utility formulation marginal utility of each product is independent of
the consumption of other products. In a more general case the summation sign is usually
raised to power
1
.

 is restricted to fall between 0 and 1.   1 to satisfy diminishing marginal
utility. And   0 to allow for zero consumption of some varieties. These restrictions
imply that the elasticity of substitution is greater than 1:
q < 1Þ
s -1
< 1Þ s -1 < s Þ -1 < 0
s
 0
 1
 0   1  0    1

Some useful transformations:
q=
s -1
1
, s=
,
s
1- q
s -1 =
q
1- q
.
Every producer i produces a unique product that requires the following amount of labor
li     xi
The shipped good is used up in the transportation, so that only the portion g of the
shipped good arrives at the destination. With transportation cost there is a difference
between the factory and the delivered price because the buyer pays for the delivered good
and for the melted part. Denote delivered price with a hat to get the price of the domestic
product in the foreign country and the price of the foreign product at home
pˆ 
p
g
and
pˆ  
p
g
From the first order conditions, the relative demand for foreign varieties at home is:
2
1
cˆ  p 1
 
c  pˆ  
Quick reminder: By consumer first order condition the ratio of marginal utilities equals
prices
U

cˆ  pˆ
U
p
c
or
 cˆ 1 pˆ 

 c 1
p
Which gives
1
cˆ  p 1
 
c  pˆ  
Introducing transportation cost yields
1
ĉ  p 1

g
c  p 
Using definition of prices we have the expression for relative demands in terms of
relative wage
1
ĉ  w 1
 g
c  w 
Demands
Set income equal to expenditures on all goods
Lw = å pi ci
i
Express value of every good relative to good 0
Lw = p0c0 å
i
pi ci
p0c0
q
pi ci æ pi æq -1
=
æ
p̂0c0 æ
æ p0 æ
3
q
q -1
æp æ
Lw = p0c0 æ æ i æ
i æ p0 æ
q
1
= p01-q c0 æ piq -1
i
Rearranging the above expression gives
-
1
c0 = p0 1-q
Lw
q
å piq -1
i
It is useful to rewrite the above expression as
-
c0 p0 = p0
q
1-q
Lw
-
q
(1-q ) æ 1-q
æ
q
q
æ
æ
æ
æ
q -1
p
æ
i
æ
ææ
æ
æi
æ
æ
æ
æ
æ
And then simplify it to get the expression for the share of income spent on each product
-
q
æ
æ 1-q
æ
æ
æ
c0 p0 æ
p0
=æ
æ
1-q
Lw æ
q
q
æ
æ æ
æ æ p 1-q
æ
i
æ
æ æ
æ
æ æ
ææ i
The term in the denominator on the right hand side is the CES price index
-
q
æ
æ
1-q
P = ææ pi æ
æi
æ
1-q
1
q
or
æ
æ1-s
P = ææ pi1-s æ .
æi
æ
So the CES demand is a function of the product price relative to the overall price level.
The higher is the overall price level the higher is the demand. Several properties of the
price index are of note:
(1) even when the prices are the same, the number of varieties makes the price index
smaller if s > 1:
1
P = n1-s p̂,
p̂i = p̂"i.
(2) the elasticity of substitution governs how much the number of goods affects the
price index. When the elasticity of substitution is high, the goods are closer
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substitutes, the power is the smaller negative number, the price index is larger, ,
the demand is lower.
Substituting elasticities of substitution for the preference parameters  the expression
becomes
æ
æ
c0 p0 æ
p0
=æ
1
Lw æ
1-s
æ
æ
1-s
æææ pi æ
æ
æ
æ i
1-s
æ
æ
æ
æ
æ
æ
æ
1-s
æp æ
=æ 0æ
æPæ
Several interesting properties of the above expression. Use the demand to construct the
indirect utility. Note that here we use the formulation where the summation sign is raised
to power 1 .
q
1
q q
1
æ æ
æ
æ
1q
æ
æ ææ
ææ
æ ææ
æ
ææ
pj
æ ææ
æ
æ
U = ææ æ
Lwæ æ
æ
1q
(
)
æ
æ
q
æ
æ j æææ
æ
q -1
ææ
p
æ
i
æ
æ æææ
æ
ææ
æ
æ
æ æææ i
ææ
æ
æ
Notice how the whole expression is raised to the power 1/ 
U = Lw
æ
ææ
æi
æ æ
1
ææ p j
q
ææ j æ
æ
1-q
pi æ
æ
( )
1
-
q
1-q
ææq
ææ
ææ
Further simplify to get
U=
Lw
-
q
æ
æ
1-q
p
ææ i æ
æi
æ
1-q
q
=
Lw
P
Utility can be thought of as the real wage where the price level captures changes in real
wages. Noting that if we define utility as the CES quantity index
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s
1
æ
æq æ æ s -1 ææs -1
U = Q = ææ cqj æ = ææ æc j s ææ
ææ
æ j
æ æ j æ
( )
the total income (or total expenditure) at home can be expressed as the
Lw = PQ
Assume that all products i are equally priced the utility can be expressed as a function of
the number of products
1-q
Lw
U=
-
æ -1-qq æ
ænp æ
æ
æ
1-q
=nq
q
1
Lw
Lw
= ns -1
p
p
Note the role of the elasticity of substitution  . It also reflects the degree of love for
variety. Larger  moves the products closer to perfect substitutes and reduces n
1
 1
by
moving it closer to 1. That is any number of varieties would give the same utility as any
one of them, perfect substitutes.
Pricing
Krugman follows Dixit, Stiglitz’s (1977) assumption of monopolistic competition and
ignores strategic interaction between the firms. Each firm effectively disregards its effect
on the combined price index. Firm’s profit is given by
 j  p j x j   w   wxi
p
 w  w

 1 
Note that constant markup also implies that the variable profits of the firm
p = x( p - b w)
Are constant share of the total revenue
p = x( p - pq ) = xp(1- q )
or
p
xp
= (1- q ) =
1
s
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This result is useful because it implies that for a given fixed cost the firm that can collect
higher revenue is also the firm that will have higher profit.
Free entry
Firms enter until their profits are equal to zero
  px   w   wx  0
Substituting for the price gives us
w
x   w   wx  0

So each firm produces
x
   
   1
  1    
Larger fixed cost leads to higher output per firm, and so does smaller variable cost.
Higher elasticity of demand means that markups are smaller and to cover a given fixed
cost firm will have to produce higher quantity.
Number of varieties.
After determining the output per firm we can determine the amount of labor that each
firm consumes:
l   x
or
l   

  1  

This give us the number of varieties that a country can produce in equilibrium
n
L

and
n* 
L*

Higher fixed cost implies fewer firms. Higher love for variety (smaller  ) also means
more firms.
World trade in the frictionless world.
7
In the absence of transportation cost each country will produce the same number of
varieties. This means that each consumer will have access to more varieties. The wage in
terms of any product remains the same because the price is still determined by the
monopoly pricing but the real wage increases because it is a function of the number of
varieties.
Why larger country has higher wage in equilibrium.
It is useful to define relative total demand (in quantity terms) for a representative
imported versus a representative local variety. Note that for ĉ* units of the foreign
variety to be consumed by the domestic consumers the foreign producers need to produce
ĉ* / g to account for the part that melts in the way:
ĉ* / g
r=
c
Using the expressions for relative demands, for the home country the expression above is
1
1
q
æ p æ1-q 1-qq
1-q 1-q
r = æ *æ g =w g
æp æ
Similarly for the foreign country the expression is
-
æ pæ
r =æ *æ
æp æ
*
1
1-q
q
g
1-q
=w
-
1
1-q
q
g
1-q
Notice that good that uses relatively cheaper labor commands a higher total expenditure
than the good that uses relatively more expensive labor which translates into:
(1)

0

 
0

From the budget constraint at home we get
8
Lw = pnc + p̂* n*ĉ*
æ pc
æ
æ p c
æ
= p̂*ĉ* æn * * + n* æ = p̂*ĉ* æn * * + n* æ
æ p̂ ĉ
æ
æ p̂ ĉ
æ
æ w 1
æ
æ w
æ
= p̂*ĉ* æn *
+ n* æ = p̂*ĉ* æn + n* æ
æ r
æ
æ w / g rg
æ
The demand for the foreign variety in the home country can be expressed in value terms
relative to the foreign wage as
p̂*ĉ*
r Lw
=
*
w
nw + r n*
This is the value of home country’s imports of each foreign variety.
Following a similar procedure we can derive the foreign demand for the domestic
varieties in value terms, which will give us the value of home country’s exports. Start
with the foreign budget constraint
L* w* = p̂nĉ + p* n*c*
æ
p*c* * æ
= p̂ĉ æn+
n
p̂ĉ æ
æ
æ
(
= p̂ĉ n+ w -1r *-1n*
)
So the demand in value terms is
p̂ĉ
r * L*w
=
w* r *nw + n*
The trade balance of the home country in terms of foreign wage is therefore:
B
r *nw
r n*
*
=
LLw
w* r *nw + n*
nw + r n*
Using the expression for the number of varieties produced in each country n 
L 1   

the expression can be further simplified to
æL* (1- q ) æ
æL (1- q ) æ
r
L
w
æ
æw
æ
B
æ a æ
æ a
æ
=
*
æL* (1- q ) æ æL (1- q ) æ
æL* (1- q ) æ
æL (1- q ) æ
w
r* æ
w
+
w
+
r
æ
æ
æ
æ
æ
æ
æ
æ a æ
æ a
æ æ a æ
æ a
æ
r * L* æ
9
Then to
B
r * L* Lw
r LL*w
=
w* r * Lw + L* w L + r L*
Expressing labor endowment in relative terms and dividing both sides by L we get
B
=
Lw*
(2)
r *w
r*
L
w +1
L*
-
rw
L
w+r
L*
Rewrite
B
1
1
=
*
L
1
Lw
w Lw
w+ *
+1
*
L
r
L* r
The first term is a scaled version of the export value and the second term is a scaled
version of the import value. We know that the relative share  increases and  
decreases with an increase in relative wage. The first term is unambiguously decreasing
in w , the sign of the second term depends on the effect of w on w / r . Using the
definition of r we get w / r = w 1-s g1-s , which is decreasing in w because s > 1. The
second term is therefore also decreasing in w . This gives us that
¶B
< 0.
¶w
So, if we start with equal wages, and sign the trade balance condition, we will know
which way the wage needs to adjust to restore equilibrium. Start with equality of wages
and show by contradiction that it is not an equilibrium. In case   1 we know that
     1 and the expression for the trade balance becomes
10
B
1
1



L
L
 Lw   1

L
L
L
L
     1

L
 L
 L
 L

    1    
 L
 L

L

   1 1   
L

 
L

 L

    1    
 L
 L

So the trade balance condition for the case of equal wages,   1 , becomes:
L

B  0     1 1     0
L

The case of   1 is familiar. It is the case of no transportation cost. When there is no
transportation cost any amount of the final good that is shipped arrives completely, g  1 ,
so that   1 . When transportation costs are present r < 1, trade is not balanced if the
countries differ in size. For convenience, think that Home is the larger country: L  L .
Then trade balance is positive at equal wages. To realize that the relative wages adjust
upwards to the equilibrium, note that the balance of trade is decreasing in relative wage.
Both terms in the expression (2) for the trade balance (the value of exports and the value
of imports) are decreasing in relative wage. To see this rewrite (2) as
B
w
w
=
*
L 1
1 L
Lw
+ *
+1
*
r L*
L r
Applying inequalities (1) yields:
 B 
  
 Lw   0

So the relative wage would have to adjust upwards from one to bring the trade balance
back to zero. Hence, in equilibrium the larger country has a larger wage.
11
The home market effect
A useful way to rewrite equation (25) from the paper in terms of shares.
L *-r
n
= L
n* 1- r L *
L
note that
n
n
n*
=
n + n* 1+ n
n*
L *-r
æ L * - r ææ
æ
1- r L *
L
L
ææ
æ=
=æ L
L
L
L
L
æ
æ1- r L* æ
ææ
æ1- r L* + L* - r æ
æ 1- r + L* (1- r )
-r
L*
=
(1- r ) 1+ L
L
(
=-
+ 1- 1- r
1
1+ r æ
L æ
L*
=
1æ
æ
æ
L + L* æ
(1- r ) 1+ L L* 1- r 1- r
L
L*
)
=
(
)
r
æ1+ r ææ L æ
+æ
æ
æ
1- r æ1- r æ
ææ L + L* æ
To check intuition, note that when endowments are equal the expression becomes:
æ1+ r æ 1 -2 r +1+ r 1
n
r
=+æ
=
= .
*
n+ n
1- r æ1- r æ
2 (1- r )
2
æ2
12