Monopolistic Competition in Trade
• The Dixit-Stiglitz Model
• Monopolistic Competition and Trade in a One-Sector Model
• Monopolistic Competition and Trade in a Two-Sector HO
Model
• Transport Costs and the Home-Market Effect
• Economic Geography
© J.P. Neary Tuesday, 07 April 2015
1
Monopolistic Competition
Due to Chamberlin; key features:
1. Differentiated products: reflecting a “taste for variety”
– Hotelling approach (used by Helpman, JIE 1981): each
consumer has an “ideal type” - difficult!
– Dixit-Stiglitz “taste for variety” approach is now standard
– Both approaches have identical implications for positive
questions (but not for normative ones)
2. Increasing returns (due to fixed costs perhaps)
– Otherwise, every conceivable variety could always be
produced, in tiny amounts
3. Free Entry => No long-run profits
– Just like perfect competition
4. No strategic behaviour: Firms ignore their interdependence
when taking their decisions.
2
1. Tastes: Dixit-Stiglitz Utility Function
n
•
•
•
•
•
q
u  (  i 1 x i )
1/q
A symmetric CES function
xi is the consumption of variety i
n, the number of varieties, is given to consumers, but endogenous in equilibrium
The index q is a measure of substitutability, and must lie in [0,1]
s
As we will show, it is related to the elasticity of substitution s : s  q  0 q  s 
{0 <q < 1} <=> {1 < s < }
1
1
1
A. Preference for Variety/Diversity
Proof: Assume all varieties have the same price p and are
consumed in equal amounts, so total expenditure is I = npx
xi  x

q
u  ( nx )
1/ q

n
1/ q
x
 n
1 / ( s 1 )
I / p
(This is the indirect utility function in symmetric equilibria)
u 
n
Logarithmically differentiating, with I and p fixed:
i.e., utility rises with diversity, and by more so the lower is s
1
s 1
QED
3
The indirect utility function can be inverted to get the
expenditure function in symmetric equilibria:
e ( p , u )  Pu
w here:
P  pn
1 / ( 1 s )
The unit expenditure function P is a true price index for the
industry.
It is decreasing in n (again, because consumers like variety)
and to a greater extent the lower is s
1.00
s
s
s
s
0.90
0.80
0.70
P
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0
10
20
30
40
50
n
60
70
80
90
100
4
B. Derive the Elasticity of Substitution
Rewrite utility function as uq and differentiate w.r.t. pi:
(q u
q 1
)ui  q x i
q 1
= pi

Now, take ratio for goods i and j:
xi  pi 
 
xj  pj
(from consumer’s FOC)
s
s 
1
1 q
0
q

s 1
s

{0<q<1} <=> {1<s<}
C. Marshallian Demands
Solve for xj, multiply by pj, sum over j and substitute into the
budget constraint, S pjxj = I, to obtain:
p
xi 
S
s
i
1 s
jpj
I
D. Industry Price Index
Substitute into u(x) to get the indirect utility function V(p,I):
q
q
u  S ixi 
So:
s q
S
1 s
j
pj
I  S j p j
q
S i pi

q
u  S j p j
1 s
1 s


1 q
I
q
1/ ( s 1 )
I
5
Finally, as in the symmetric case, this can be inverted to get the
expenditure function:
e( p , u )  P u
w here:
P  S j p j
1 s

1 / ( 1 s )
The demand functions can be expressed more simply in terms
of the unit expenditure function P :
 pi 
xi  

 P
s
I
P
;
 pi 
c
xi  

 P
s
u
(The Marshallian demand function is log-linear in relative price and real income, both defined
with respect to P)
6
Production: Profit Maximisation by Firms
Firms maximise profits given the (perceived) demand curve (i.e.,
they take income and the industry price index as given):
1 / s
p i  Ax i
w here:
A P
( s 1 )/ s
I
1/ s
Hence their total and marginal revenue curves are (suppressing i):
TR  Ax
( s 1 )/ s
and
MR 
s 1
s
Ax
1 / s
 qp
So the demand and MR curves are iso-elastic, with the latter a
fraction q of the former.
Costs
Homotheticity: Production uses a composite input, at unit cost W
Overheads require F units; and production c units per unit output:
TC = (F+cx)W
(Set W=1 for now)
Hence:
MC = c
AC = c+F/x
(a rectangular hyperbola)
7
Dixit-Stiglitz (symmetric CES) preferences:
p
-1/s
==> Demand: p = Ax
Marginal Revenue: q p
D
MR
x
8
Total Costs: C = F+cx
p
==> Marginal Costs: c
Average Costs: c+F/x
AC
c
MC
x
9
Profit Maximisation
MC=MR => c = q p
i.e., p = c /q (p independent of A, F)
Alternatively: p /c = 1/q = s /(s -1) (the price-cost margin
is decreasing in s)
Free Entry
AC=AR => c + F/x = p
+ {MC=MR}
In equilibrium:
=> c + F/x = c / q
p  Ax
 1/ s
=>

x = (s -1) F/c
A ( F , c,s )  p x
1/ s
i.e., Equilibrium A is also independent of P, I, and therefore n
1
p
Equilibrium: Chamberlinian Tangency
• Profit maximisation: MR=MC
• Free entry: p=0
==>
• Firm output depends only on F, c, s
• Industry output adjusts to demand
shocks via changes in n only
c/q
D
c
AC
MC
MR
sF/c
x
1
Technical Digression
As drawn, the AC curve is more convex than the demand curve.
Proof that this must be so:
p  Ax
 1/s


AC  c 
p  
p  

F
x


A C   p   2 F s
2
1
s
s 1
2
s
Ax
Ax

s 1
s

2 s 1
s
AC   
A C  

s 1 p
2
2
s
x
F
x
2
2F
x
3
 ( s  1) p x
 F s [ 2 s  ( s  1)]
( u sin g p x  s F )
 F s ( s  1)
So, s > 1 is necessary and sufficient for AC to be more convex.
1
Equilibrium Firm Size
In equilibrium, x depends only on c, F and s: all adjustment to
changes in other exogenous variables is via changes in n
How to avoid this implausible property?
1. Relax CES assumption.
[Krugman, JIE 1979]
2. Assume more than one factor with non-homothetic costs:
[Lawrence/Spiller, QJE 1983; Flam/Helpman, JIE 1987; Forslid/Ottaviano, JEG 2003]
TC = rf + wax  x = (s -1)rf/wa
3. Assume heterogeneous firms:
[Melitz, Em 2003]
xi = (s -1)Fi/ci
1
Role of s
High s:
• different varieties are close substitutes for each other
(preference for diversity is not so strong)
• p close to c : so p and MR curves are flat and close together
• x large: economies of scale are highly exploited
• fewer varieties, higher output of each
Low s:
• different varieties are less close substitutes
(greater preference for diversity)
• p >> c : so p and MR curves are steep and far apart
• x small: economies of scale are not highly exploited
• more varieties, lower output of each
1
p
B
A
AC
MC
MR
(low s)
MR
(high s)
x
Effects of Changes in the Elasticity of Substitution
1
Monopolistic Competition and Trade in a One-Sector Model
[Krugman, JIE 1979]
2 countries; 1 sector; labour the only factor of production
(W=w=1); identical technology and tastes in both countries.
Full employment: L = nLi = n(F+cx) => n = L /(F+cx)
But:
x  ( s  1)
F
c

n 
L
Fs
i.e., number of varieties is linear in the size of the economy
Autarky: L, L* => nA , nA*, x
Trade:
L  L
• All trade is intra-industry
• Trade is unrelated to comparative advantage
L*

n
(Both countries have the same autarky prices, since they are identical
except for size, which has no effect given identical homothetic tastes)
• Trade is welfare improving (since it increases the number of varieties available)
• Volume of trade is maximised when countries are of equal size
1
Monopolistic Competition and Trade in the Two-Sector
Heckscher-Ohlin Model
We extend the two-sector HO model by assuming that one sector
has a Dixit-Stiglitz monopolistically competitive structure.
2 sectors:
• X1 “Food”: perfectly competitive, output homogeneous, p=1
• X2 “Manufactures”: monopolistically competitive
Tastes:

U  X1 X
1 
2
X 2  (  i 1 xi )
n
,
q
1/q
An example of “two-stage budgeting”:
• the utility function is Cobb-Douglas in food and manufacturing;
• the manufacturing sub-utility function in turn has the Dixit-Stiglitz form.
Expenditure function:
e (. )  P
1 
u
w here:
P  [ np
1 s
n (p )
*
*
1 s
1/ ( 1 s )
]
1
Economic Geography
Standard features of Dixit-Stiglitz type models:
1. Demand intercept A depends:
• positively on industry price index P
• positively on expenditure I
2. Industry price index depends:
• negatively on number of firms at home and abroad
• positively on trade costs
Additional feature of Venables model: Each firm uses the output
of every other as an input. So:
• Expenditure I depends positively on n
• Input costs depend positively on P
[Now we need to make W explicit: replace F and c by FW and cW]
Implications for stability of diversified equilibria:
1
p
Effects of entry by one new firm:
1. P, P* fall => p fall
2. Cost linkage:
P falls => W falls => p rise
3. Demand linkage:
Demand rises => p rise
3
cW/q
1
cW
AC
D
MC
MR
sF/c
2
x
1

1
Core
Dispersal
0.5
Periphery
TB
TS
T
Figure 2: Agglomerated and Dispersed Equilibria
as a Function of Trade Costs
2
Heterogeneous Firms
Firm heterogeneity in monopolistic competition:
[Melitz (Em 2003), Helpman-Melitz-Yeaple (AER 2004)]
• Firms pay a sunk cost to reveal their productivity
Draw c from g(c) with positive support over (0,)
• Given their productivity, they calculate their expected profits
and choose to produce or exit
Exit if c < ce where p(ce) = 0 or r(ce) = f .
• If exporting and/or FDI require an additional fixed cost, only
high-productivity firms will engage in them
• Predictions are consistent with micro-empirical evidence
2