Intra-Industry Trade
• The Krugman model
• n varieties of the same good exist
• labour L is the only factor of production
li = α + β xi
• li is the amount of labour used to produce
a quantity x of the good (variety) i; α is
fixed cost; β is marginal cost
n
• The utility function U = ∑ v ( ci )
i=1
represents consumers preferences. All n
varieties of the good enter the utility
function. (Dixit-Stiglitz approach)
• Full employment is assumed, therefore
n
n
i=1
i=1
L = ∑ li = ∑ [α + β xi ]
• In equilibrium, demand for each variety is
equal to supply
xi = Lci
• The model must be solved for three variables:
• output for each good (variety) xi
p
• the ratio
w
• (w is determined in a competitive market)
• the number (variety) of goods n
Consumers equilibrium
• Consumers maximize the utility function
n
U = ∑ v ( ci )
i=1
• subject to the budget constraint
n
w = ∑ pi ci
i=1
• The Lagrangian to maximize is
n
⎡
⎤
 = ∑ v ( ci ) + λ ⎢ w − ∑ pi ci ⎥
⎣
⎦
i=1
i=1
n
• Maximizing the Lagrangian with respect to ci
we get the first order condition (FOC)
⎡
⎤
 = ∑ v ( ci ) + λ ⎢ w − ∑ pi ci ⎥
⎣
⎦
i=1
i=1
δ
= v′ ( ci ) − λ pi = 0
δ ci
v′ ( ci ) = λ pi
n
• Combining FOC with
n
xi
xi = Lci → ci =
L
• we obtain the inverse demand function pi =
v′ (
xi
λ
L
)
Firms
• Firms maximize the profit function:
π = pi xi − wli
pi =
v′ (
xi
λ
L
)→
π=
v′ (
xi
λ
L
)x
i
− w (α + β xi )
• Maximizing with respect to output xi :
1
xi
xi
v
x
+
v
′′
′
(
(
L)
L)
i
dπ
L
=
− wβ = 0
dxi
λ
• Simplifying notation, the FOC becomes
v′′ci + v′
= wβ
λ
• or, after some algebraic manipulation, recalling
that
xi
v′ ( L ) v′
pi =
=
λ
λ
v′ ⎛ v′′ci ⎞
+ 1⎟ = wβ
⎜⎝
⎠
λ v′
⎛ v′′ci ⎞
pi ⎜
+ 1⎟ = wβ
⎝ v′
⎠
• Defining
• then,
v′
e=−
v′′ci
⎛ v′′ci ⎞
pi ⎜
+ 1⎟ = wβ
⎝ v′
⎠
may be written as
⎛ 1⎞
p ⎜ 1 − ⎟ = wβ
⎝
e⎠
[ marginal cost ]
p
e
=
β
w e −1
• e is the income elasticity of demand
• when π = 0, market is in equilibrium, otherwise
• if π > 0 new firms enter the market
• if π < 0 firms exit the market
• the above equilibrium condition implies that
px − w (α + β x ) = 0
• or
p
α
α
=β+ =β+
w
x
Lc
Autarkic equilibrium
• PP line
• ZZ line
p
e
=
β
w e −1
p
α
α
=β+ =β+
w
x
Lc
• Intersection of the two lines determines
equilibrium consumption c and as a
consequence the number n of goods (variety)
that are produces
L
n=
li
li = α + β xi
L
n=
α + βx
xi = Lci
L
n=
α + β Lc
International equilibrium
• There are no transportation costs
• Symmetry among countries implies that wages
and prices are equal at the international level
• International trade increases firms dimension
• The variety of goods available for consumption
increases
• Direction of trade flows in not determined
• Trade volumes are determined
International equilibrium
• With international trade, utility function is
n
U = ∑ v ( ci ) +
i=1
n+n*
∑ v (c )
i
i=n+1
• The number of goods produced in the two
countries is proportional to labour endowment
L
n=
α + βx
L*
n* =
α + βx
International equilibrium
• The share of goods produced in the two
countries in the world economy are
*
*
n
L
n
L
=
;
=
*
*
*
*
n+n
L+L n+n
L+L
• We have balanced trade when value of import
of a country is equal to value of export by the
other country (wL is income of home country)
wL ⋅ L
wL ⋅ L
= M = M* =
*
*
L+L
L+L
*
*
• We may determine trade volume but not trade
direction.