1 Monopolistic Competition in Trade The Dixit-Stiglitz Model Monopolistic Competition and Trade in a...
Transcript of 1 Monopolistic Competition in Trade The Dixit-Stiglitz Model Monopolistic Competition and Trade in a...
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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 Wednesday 19 April 2023
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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.
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1. Tastes: Dixit-Stiglitz Utility Function
• 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 is a measure of substitutability, and must lie in [0,1]
• As we will show, it is related to the elasticity of substitution :{0 << 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
(This is the indirect utility function in symmetric equilibria)
Logarithmically differentiating, with I and p fixed:
i.e., utility rises with diversity, and by more so the lower is QED
u n 1
1
u xii
n ( ) / 1
1
x x u nx n x n I pi ( ) // / /( ) 1 1 1 1
1
110
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The indirect utility function can be inverted to get the expenditure function in symmetric equilibria:
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
e p u Pu P pn( , ) /( ) where: 1 1
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 10 20 30 40 50 60 70 80 90 100
n
P
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B. Derive the Elasticity of Substitution
Rewrite utility function as u and differentiate w.r.t. pi:
(from consumer’s FOC)
Now, take ratio for goods i and j:
{0<<1} <=> {1<<
C. Marshallian Demands
Solve for xj, multiply by pj, sum over j and substitute into the budget constraint, pjxj = I, to obtain:
D. Industry Price Index
Substitute into u(x) to get the indirect utility function V(p,I):
So:
( ) u u x pi i i
1 1 =
xx
pp
i
j
i
j
1
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x Iip
pi
j j
1
u x I p Ii ip
p j ji i
j j
1
1 1
u p Ij j 1 1 1 /( )
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Finally, as in the symmetric case, this can be inverted to get the expenditure function:
The demand functions can be expressed more simply in terms of the unit expenditure function P :
(The Marshallian demand function is log-linear in relative price and real income, both defined
with respect to P)
e p u Pu P pj j( , )/( )
where: 1 1 1
xp
P
I
Px
p
Pui
iic i
;
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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):
p Ax A P Ii i 1 1 1/ ( )/ / where:
TR Ax MR Ax p ( )/ /
1 1 1and
Hence their total and marginal revenue curves are (suppressing i):
So the demand and MR curves are iso-elastic, with the latter a fraction of the former.
CostsHomotheticity: 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)
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MR
p
x
D
Dixit-Stiglitz (symmetric CES) preferences:
==> Demand: p = Ax Marginal Revenue: p
-1/
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p
x
MC
AC
Total Costs: C = F+cx
==> Marginal Costs: c Average Costs: c+F/x
c
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Profit Maximisation
MC=MR => c =p i.e., p = c / (p independent of A, F)
Alternatively: p /c = 1/ = /( -1) (the price-cost margin is decreasing in )
Free Entry
AC=AR => c + F/x = p
+ {MC=MR} => c + F/x = c / => x = (-1) F/c
In equilibrium: p Ax A F c px 1/ 1/ ( , , )
i.e., Equilibrium A is also independent of P, I, and therefore n
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MR
p
F/c
c
c/
x
MC
ACD
Equilibrium: Chamberlinian Tangency• Profit maximisation: MR=MC • Free entry: =0 ==>
• Firm output depends only on F, c, • Industry output adjusts to demand shocks via changes in n only
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Technical Digression
As drawn, the AC curve is more convex than the demand curve.
Proof that this must be so:
p Ax p Ax
p Axp
x
1 1
1 1
1
2
2 1
2 2
/
AC c AC
AC
Fx
Fx
Fx
2
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AC p F px
F px F
F
2 1
2 1
1
2
( )
[ ( )] ( )
( )
using
So, > 1 is necessary and sufficient for AC to be more convex.
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Equilibrium Firm Size
In equilibrium, x depends only on c, F and : 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 + wx x = (-1)rf/w3. Assume heterogeneous firms:
[Melitz, Em 2003]
xi = (-1)Fi/ci
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Role of
High :
• 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 :
• 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
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MR(high )
p
x
MC
AC
B
Effects of Changes in the Elasticity of Substitution
MR(low )
A
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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:
i.e., number of varieties is linear in the size of the economy
Autarky: L, L* => nA , nA*, xTrade: • All trade is intra-industry• Trade is unrelated to comparative advantage
(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
x nLF
Fc ( )
1
L L L n *
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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:
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:
U X X X xii
n
1 21
2 1
1 , ( ) /
e P u P np n p(.) [ ( ) ]* * /( ) 1 1 1 1 1 where:
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Economic GeographyStandard 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:
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MR
p
F/c
cW
cW/
x
MC
ACD
Effects of entry by one new firm: 1. P, P* fall => fall 2. Cost linkage: P falls => W falls => rise 3. Demand linkage: Demand rises => rise
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2
1
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T
Dispersal
Figure 2: Agglomerated and Dispersed Equilibriaas a Function of Trade Costs
Core
TSTB
Periphery
0.5
1
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Heterogeneous Firms
Firm heterogeneity in monopolistic competition:[Melitz (Em 2003), Helpman-Melitz-Yeaple (AER 2004)]
• Firms pay a sunk cost to reveal their productivityDraw 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 (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