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1. THE CORE-PERIPHERY MODEL:
KEY FEATURES AND EFFECTS*
Richard Baldwin, Rikard Forslid, Philippe Martin, Gianmarco Ottaviano and Frederic
The basic structure of the core-periphery (CP) model is astoundingly familiar to
trade economists. Take the classroom Dixit-Stigliz monopolistic-competition trade model
with trade costs, add in migration driven by real wage differences, impose a handful of
confusing normalisations, and voila, the CP model! The fascination of the CP model
stems in no small part from the fact that these seemingly innocuous changes so
unexpectedly and so radically transform the behaviour of a model that trade theorists
have been exercising for more than 25 years. 1
This chapter presents the core-periphery model, or more precisely the version in
Chapter 5 of Krugman, Fujita and Venables (1999) – FKV for short. We describe all the
main properties of the model including catastrophe, hysteresis, and global stability. The
intention is to provide a definitive list of such properties in order to provide a standard of
comparison for other economic geography models.
1.1. The Standard Core-Periphery Model
The basic structure of the core-periphery (CP) model is shown schematically in
Figure 1.1. The model assumes two initially symmetric regions (north and south), two
factors (industrial workers H and agricultural labourers L), and two sectors (manufactures
M and agriculture A).2 The M-sector is a standard Dixit-Stiglitz monopolistic competition
sector, where manufacturing firms (M-firms for short) employ H to produce output
subject to increasing returns. In particular, production of each M-variety requires a fixed
cost of F units of H in addition to am units of H per unit of output, so the cost function is
w(F+amx), where x is a firm's output and w is the reward to H. The A-sector produces a
homogeneous good under perfect competition and constant returns using only L.
Both M and A are traded, with A trade being frictionless while M trade is
inhibited by iceberg trade costs.3 Specifically, it is costless to ship M-goods to local
consumers but to sell one unit in the other region, an M-firm must ship τ≥1 units. The
* This is a draft chapter of a manuscript "Public Policies and Economic Geography" by Richard Baldwin,
Rikard Forslid, Philippe Martin, Gianmarco Ottaviano and Frederic Robert-Nicoud.
1 Its first use in the trade literature was Victor Norman’s 1976 unpublished paper (the substance of which
appears in Dixit and Norman 1980).
2 FKV use LM and LA for, respectively, the mobile factor (our H) and the immobile factor (our L). Our
notational choice is motivated by the fact that most of this book deals with issues where labour, or at least
unskilled labour, is the immobile factor and physical, human and/or knowledge capital is the mobile factor.
3 See Krugman, Fujita and Venables (1999 Chapter 7) and Davis (1999) for how variations on this
assumption affect outcomes.
idea is that τ-1 units of the good “melt” in transit. As usual, τ captures all the costs of
selling to distant markets, not just transport costs, and τ-1 is the tariff-equivalent of these
Figure 1.1: Schematic Diagram of the CP Model
The representative consumer in each region has preferences consisting of CES
preferences over M-varieties nested in a Cobb-Douglas upper-tier function that also
includes consumption of the homogenous good, A. Specifically:
1 c C CCC CUnn
where CM and CA are, respectively, consumption of the M composite and consumption of
A. Also, n and n* are the number (mass) of north and south varieties, µ is the expenditure
share on M-varieties, and σ is the constant elasticity of substitution between M-varieties.
Regional supplies of L as well as the global supply of H are fixed (at Lw and Hw,
respectively), but the inter-regional distribution of H is endogenous with H flowing to the
region with the highest real wage. As in FKV, migration is governed by the ad hoc
- Dixit-Stiglitz monopolistic competition
- Increasing returns:
- Fixed cost = F units of H
- variable cost = am of H per unit output
H migration driven by real wage difference,
ω−ω∗ ≡ w/P - w*/P*
costs for M
-Walrasian (CRS & Perf.Comp.)
-variable cost = aA units L per unit A
-A is numeraire (pA=1=wA)
where sH is the share of world H in the north, H is the northern labour supply, Hw is the
world labour supply, ω and ω* are the northern and southern real wages, w is the
northern nominal wage, and P is the north’s region-specific perfect price index implied
by (1.1); pA is the price of A and pi is the price of M-variety i (the variety subscript is
dropped were clarity permits). Analogous definitions hold for southern variables, which
are denoted with an asterisk.
1.1.1. Equilibrium Expressions
As is well known (see Appendix 1), utility optimisation yields a constant division
of expenditure between M and A, and CES demand functions for M varieties, namely:
where E is region-specific expenditure, w is the wage for H and wL is the wage rate of L.
As usual in the Dixit-Stiglitz monopolistic competition setting, free and instantaneous
entry drives pure profits to zero, so E involves only factor payments. Demand for A is
On the supply side, perfect competition in the A-sector forces marginal cost
pricing, i.e. pA=aAwL and pA*=aAwL*. Costless trade in A equalises northern and southern
prices and thus indirectly equalises L wage rates internationally, viz. wL=wL*. In the M-
sector, ‘milling pricing’ is optimal (see appendix 1), so the ratio of the price of a northern
variety in its local and export markets is just τ. Summarising these equilibrium-pricing
results, we have:
where p and p* are the local and export prices of a home-based M-firm. Analogous
pricing rules hold for southern M-firms.
A well known result for the Dixit-Stiglitz monopolistic competition model is that
operating profit (call this π) is the value of sales divided by σ, where the value of sales is
either shipments at producer prices, or retail sales at consumer prices.4 Using milling
pricing from (1.3) and the shipments-based expression for operating profit in the free
entry condition, namely px/σ=wF, yields the equilibrium firm size. This and the full
employment of H – i.e. n(F+amx)=H – yields the equilibrium number of firms, n.
Specifically, the equilibrium number and scale of firms are:
4 A typical first order condition for local sales is pi(1-1/σ)=waM. Rearranging this, operating profit on local
sales is (p-waM)ci=pc/σ. A similar rearrangement of the first order condition for export sales and
summation yields the result for consumer prices. Noting that p*c*=wτc*=wτxhh/τ, where xhh is export
shipments, yields the result for producer prices.
is the equilibrium size of a typical X-firm. Similar expressions define the
analogous southern variables. Two features of (1.5) are worth highlighting. First, the
number of varieties produced in a region is proportional to the regional labour force. H
migration is therefore tantamount to industrial delocation and vice versa. Second, the
scale of firms is invariant to everything except the elasticity of substitution and the size of
fixed costs. Note also that one measure of scale, namely the ratio of average cost to
marginal cost, depends only on σ.5
The market for northern X-varieties must clear at all moments and from (1.5) firm
output is fixed, so using (1.3), the market clearing condition for a typical Northern variety
where R is a mnemonic for ‘retail sales’. Due to markup price and iceberg trade costs, the
value of a typical firm’s retail sales at consumer prices always equals its revenue at
producer prices; R is thus also a mnemonic for revenue. Also, φ=τ1-σ measures the ‘free-
ness’ (phi-ness) of trade and note that the free-ness of trade rises from φ=0 (with infinite
trade costs) to φ=1 with zero trade costs. Equilibrium additionally requires that the
equivalent of (1.6) for a typical southern variety holds, and that the market clearing
condition for A, namely,
holds. Walras's law permits us to drop one of the three market clearing conditions.
Traditionally, the A-sector condition is dropped, but given the complexity of the M-sector
conditions, it often proves advantageous to drop the southern M-sector clearing
Eq. (1.6) and its southern equivalent are often called the wage equations since
they can be written in terms of w, w*, H and H*. One can make some progress by
plugging (1.7) instead of the southern wage equation into (1.6), but unfortunately there is
no way to solve for the equilibrium w’s analytically since 1-σ is a non-integer power.
Numerical solutions for particular values of µ, σ and φ are easily obtained.7
5 Note that, the scale elasticity is a measure that has its limitations. For instance, if the cost function in not
homothetic in factor prices, a given scale elasticity does not coincide one-to-one with firm size. If capital is
used only in the fixed cost and labour only in the variable costs, then the scale elasticity is rF/(waxx)+1.
Even if this is constant in equilibrium, the corresponding firm size increases with the capital labour ratio.
Since trade costs can in general affect factor prices, this means that trade costs can also affect firm scale,
even in the Dixit-Stiglitz model. See Flam and Helpman (1987) for details.
6 Local sales of a northern variety are w1-σ/[nw1-σ+n*(τw*)1-σ] times µE since the price of imports is τw*.
The expression for export sales is (τw)1-σ/[n(τw)1-σ+n*w*1-σ] times µE*.
7 A MAPLE spreadsheet that shows how to solve this model numerically can be found on
1.1.2. Choice of Numeraire and Units
Both intuition and tidiness are served by appropriate normalisation and choice of
numeraire. In particular, we take A as numeraire and choose units of A such aA=1. This
simplifies both the expressions for the price index and expenditure since it implies
pA=wL=wL*=1. In the M-sector, we measure M in units such that am=(1-1/σ), so that the
equilibrium prices become p=w and p*=τw, and the equilibrium firm size becomes
The next normalisation, which concerns F, has engender some confusion. Since we
are working with the continuum-of-varieties version of the Dixit-Stiglitz model, we
can normalise F to 1/σσ (see Box 1.1 on this point). With this, 1
x, n=H and n*=H*.
These results simplify the M-sector market-
where sH and sn are the north's shares of Hw and nw respectively, and, by
construction, w=w*=1 in the symmetric outcome. In the core-p clearing condition, (1.6).
The results that n=H and n*=H* also boost intuition by making the connection between
migration and industrial delocation crystal-clear.
We have not yet specified units for L or H. Choosing the world endowment of H,
namely Hw, such that Hw=1 is useful since it implies that the total number of varieties is
fixed at unity (i.e. nw=1) even though the production location of varieties is endogenous.
The fact that n+n*=1 is useful in manipulating expressions. For instance, instead of
Box 1.1: One Too Many Normalisations?
Many authors who have worked with the original CP model and its vertical-
linkages variant (due to Venables 1996) use two normalisations in the Dixit-Stiglitz
sector in order to tidy the equations. In particular, they set the variable cost to 1-1/σ
units of the sector-specific factor and they set the fixed cost to 1/σ units of the same
factor. Since units of the sector-specific factor are also normalised elsewhere, it may
seem that there is one too many normalisations. Peter Neary makes the point
elegantly in Neary (2000): “As Oscar Wilde's Lady Bracknell might have said, to
normalise one cost parameter may be regarded as a misfortune, to normalise both looks
The truth is that two normalisation are OK in the continuum of varieties
version of Dixit-Stiglitz, but it are not OK in the discrete version (which is the
version Neary 2000 works with).
With a continuum of varieties, n is not, strictly speaking, the number of
varieties produced in the north; indeed, as long as n is not zero, an uncountable
infinity of varieties are produced in the north. Rather “n” corresponds to a mass of
varieties that can be represented as the segment [0…n] on a real line. But what are
the units on this real line? Here we get an extra degree of freedom that can be
absorbed in an extra normalisation.
In the discrete varieties, this is not possible. With discrete varieties, firms
have a natural metric and that metric is defined by the size of the fixed cost.
writing sH for the northern share of Hw, we could write sn or simply n. Finally, it proves
convenient to have w=w*=1 in the symmetric outcome (i.e. where n=H=1/2) and core
periphery outcomes (i.e. where n=H=1 or 0). This can be accomplished by choosing units
of L such that the world endowment of the immobile factor, i.e. Lw, equals (1-µ)/2µ.8
In summary, the equilibrium values with these normalisation are:
wnnwwwppx w p w p
Note also that with these normalisations the nominal wage in the core equals unity in the
core-periphery outcomes. The nominal wage in the periphery in such outcomes varies
with trade costs. Specifically, the periphery’s wage is (φµ(1+L)+µL/φ)1/σ. Of course, this
is a sort of ‘virtual’ nominal wage, viz the wage that a small group of workers would earn
if they did work in the periphery.
1.2. The Long Run Equilibria and Local Stability
In solving for the long-run equilibria, the key variable – that is to say the state
variable – is the division of the mobile factor H between north and south.9 Inspection of
the migration equation, (1.2), shows that there are potentially two types of long run
equilibria. The first type – core-periphery outcomes – is where sH=1 or 0. The second
type– interior outcomes – is where ω=ω* but 0<sH<1. Given symmetry, it is plain that ω
does equal ω* when sH=1/2, so sH=1/2 is also always a long run equilibrium.10 It is
equally clear from the migration equation that when sH=1 or 0, the economy is also in
equilibrium since no migration occurs.
Identification of these long-run equilibria, however, is only part of the analysis.
Complete analysis requires us to evaluate the local stability of these three equilibria.
1.2.1. Local Stability Analysis
The literature relies on informal tests to find the level of trade costs where the
symmetric equilibrium becomes unstable and where the full agglomeration outcome
becomes stable. In particular, for the symmetric equilibrium, one sees how a small
northward migration changes in the real wage gap ω-ω*; if it is negative, the equilibrium
is stable, otherwise it is unstable. For the core-periphery outcomes, (CP outcome for
short), the test is whether the level of the periphery real wage exceeds that of the core; if
so, the CP equilibrium is unstable, otherwise, it is stable. Symbolically the stability tests
8 KFV takes Lw as µ and Aw as 1-µ, but wages are unity as long as Lw/Aw equals µ/(1-µ).
9 With our normalisation, we can write the state variable as n, H, sn or sH.
10 Are there other interior steady states? Numerical analysis shows that there can also be two other interior
steady states. Intriguingly, no one has been able to prove that there are no more than five steady states,
although thousands of simulations undertaken by dozens of researchers have never found more than five.
where ‘sym’ and ‘CP’ indicate evaluation at sH=½ and sH=1, respectively. The φ where
the first expression in (1.9) holds with equality is called the ‘break’ point, φB. The φ
where the second expression holds with equality is called the ‘sustain’ point, φS. The
validity of these informal tests in (1.9) is easily proved; see Box 1.2.
Box 1.2: Equivalence of Formal and Informal Local Stability Tests
Using (1.9), FKV establish that the symmetric equilibrium is stable only for
sufficiently low levels of trade free-ness, specifically for φ<φB, and that CP outcomes are
stable only for sufficiently high levels of trade free-ness, specifically for φ>φS. Using
numerical simulation, FKV also establish that there is a range of φ for which both the
symmetric and core-periphery outcomes are stable, i.e. that φS<φB.
These three facts and the long-run equilibria can be conveniently illustrated with
the so-called ‘tomahawk’ diagram, Figure 1.2 (the ‘tomahawk’ moniker comes from
The dynamic aspects of the CP model can be expressed as a single non-linear
ordinary differential equation, namely:
(1.12)*];[)1(ωω −≡ΩΩ−= HHHH ssss
where Ω[sH] relates sH to the real wage gap. Formally, local stability is evaluated by
linearising (1.12) around an equilibrium point o
s, and checking the coefficient on sH. If
it is negative, the system is locally stable, otherwise, it is locally unstable. The linear
Note that at the symmetric equilibrium (i.e. 2/1=
s) the necessary and
sufficient condition for local stability is that d(ω-ω*)/dsH<0. At the core-periphery
outcome (sH=1 or sH=0) the necessary and sufficient condition is (ω-ω*)<0. These line
up exactly with (1.9). For details, see Baldwin (2000). Evaluating the derivatives in
(1.12) at the symmetric and CP outcomes, it can be shown (more on this below) that φS
and φB satisfy:
viewing the stable-part of the symmetric equilibrium as the handle of a double-edged
axe). The diagram plots sH against the free-ness of trade, φ and shows locally stable long-
run equilibria with heavy solid lines and locally unstable long-run equilibria with heavy
dashed lines. Thus the three horizontal lines sH=1, sH=1/2 and sH=0 are steady states for
any permissible level of φ. The bowed line also represents steady states. Note that for
most levels of φ, there are three long-run equilibria, while for the levels of φ
corresponding to bowed curve, there are five equilibria – two core-periphery outcomes,
the symmetric outcome and two interior, asymmetric equilibria.11
Figure 1.2: The Tomahawk Diagram
1.3. Catastrophic Agglomeration and Locational Hysteresis
Catastrophe is the most celebrated hallmark of the CP model – probably because
it is so unexpected. Specifically, starting from a symmetric outcome and very high trade
costs, marginal increases in the level of trade free-ness φ has no impact on the location of
industry until a critical level of φ is reached. Even a tiny increase in φ beyond this point
causes a catastrophic agglomeration of industry in the sense that the only stable outcome
is that of full agglomeration.12
The key requirement for catastrophe is that the stable interior outcome becomes
locally unstable beyond a critical φ – the so-called break point – and that at the same level
of trade costs, the full agglomeration outcomes are the only stable equilibrium.
11 Of course when there is no trade φ=0 or distance has no meaning φ=1, the location of production is not
determined, so any division of Hw is a steady state.
12 In the jargon, the catastrophe property is called super-critical bifurcation.
The literature traditionally uses the tomahawk diagram to illustrate the catastrophe
feature.13 The idea is that trade costs have, roughly speaking, fallen over time. Thus
starting in the distant past – say the pre-industrial era – trade costs were very high and
economic activity was very dispersed. As time passed, φ rose, eventually to a level
beyond φB, at which point industry rapidly became agglomerated in cities and in certain
nations. Perhaps the most striking feature of the CP model is the result that a symmetric
reductions in trade costs between initially symmetric nations eventually produces
catastrophic agglomeration. That is, rising φ has no impact on the location of industry
until a critical level of openness is reached. However, even a tiny increase in φ beyond
this point results in very large location effect as the even division of industrial becomes
unstable. That non-marginal effects come from marginal changes is certainly one of the
hallmarks of the economic geography models.
The second famous feature of the CP model is hysteresis. That is, suppose we
start out with an even division of industry between the two regions and a φ between the
break and sustain points (i.e. in the so-called ‘overlap’). Given that the symmetric
outcome and both full agglomeration outcomes(core-in-the-north and core-in-the-south)
are all locally stable, some location shock, or maybe even an expectations shock, could
shift industry from the symmetry outcome to one of the core outcomes. Importantly, the
locational impact would not be reversed when the cause of the shock were removed. In
other words, this model features sunk-cost hysteresis of the type modelled by Baldwin
(1988), Baldwin and Krugman (1989) and Dixit (198?). The feature is also sometimes
called ‘path dependency’, or ‘history matters’.
The key requirement for locational hysteresis is the existence of a range of φs
where there are multiple, locally stable equilibria.
1.4. The 3 Forces: Intuition for the Break and Sustain Points
The complex equilibrium structure and extremely non-neoclassical behaviour this
model is curious, to say the least, given the fairly standard assumptions behind the model.
This section provides intuition for the complexity.
1.4.1. The Three Forces and the Impact of Trade Costs
There are three distinct forces governing stability in this model. Two of them—
demand-linked and cost-linked circular causality (also called backward and forward
linkages) – favour agglomeration, i.e. they are de-stabilising. The third – the local
competition effect (also known as the market crowding effect) – favours dispersion, i.e. it
The expressions E=wL+wAA and E*=w*(Lw-L)+wAA help illustrate the first
agglomeration force, namely demand-linked circular causality. Starting from symmetry, a
small migration from south to north would increase E and decrease E*, thus making the
northern market larger and the southern market smaller since mobile workers spend their
income locally. In the presence of trade costs, and all else being equal, firms will prefer
13 This type of application is perhaps clearest in Krugman and Venables (1995).
the big market, so this migration induced “expenditure shifting” encourages “production
shifting”. Of course, firms and industrial workers are the same thing in this model, so we
see that a small migration perturbation tends to encourage more migration via a demand-
linked circular causality.
The definition of the perfect price index in (1.2) helps illustrate the second
agglomeration force in this model, namely cost-linked circular causality, or forward
linkages. Starting from symmetry, a small migration from south to north would increase
H and thus n while decreasing in H* and n*. Since locally produced varieties attract no
trade cost, the shift in n’s would, other things equal, lower the cost of living in the north
and raise living costs in the south, thus raising the north’s relative real wage. This in turn
tends to attract more migrants.14
The lone stabilising force in the model, the so-called local competition, or market
crowding, effect, can be seen from the definition of retail sales, R, in (1.6). Perturbing the
symmetric equilibrium by moving a small mass of H northward, raises n and lowers n*.
From (1.6), we see that this tends to increase the degree of local competition in the north
and thus lower R (as long as φ<1).15 To break even, northern firms would have to pay
lower nominal wages. All else equal, this drop in w and corresponding rise in w* makes
north less attractive and thus tends to undo the initial perturbation. In other words, this is
a force for dispersion of industry activity.
The catastrophic behaviour of the model stems from two facts, which we explore
more below. The first is that the dispersion force is stronger than the agglomeration
forces at high trade costs. The second is that raising the level of trade free-ness reduces
the magnitude of each of the three forces, but it erodes the strength of the dispersion force
faster. As a result, at some level of trade costs – the break point – the agglomeration
forces become stronger than the dispersion force and industry collapses into just one
region. For readers who wish to understand these forces in more depth, we turn now to a
series of thought experiments that more precisely illustrate the forces and their
dependence on trade costs.
1.4.2. A Series of Thought Experiments
Focusing on each of the three forces separately boosts intuition and we
accomplish this via a series of thought experiments. These focus on the symmetric
equilibrium for a very pragmatic reason. In general, the CP model is astoundingly
difficult to manipulate since the nominal wages are determined by equations that cannot
be solved analytically. At the symmetric equilibrium, however, this difficulty is much
attenuated. Due to the symmetry, all effects are equal and opposite. For instance, if a
migration shock raises the northern wage, then it lowers the southern wage by the same
amount. Moreover, at the symmetric outcome, w=w*=1, so much of the intractability –
which stems largely from terms involving a nominal wage raised to a non-integer power
14 FKV call it the price index effect.
15 In Dixi-Stiglitz competition, the price-cost markup never changes, so this local competition effect is no a
pro-competitive effect. This is why some authors prefer the term “market crowding”.
The Local Competition Effect
To separate the production shifting and expenditure shifting aspects of migration,
the first thought experiment supposes that H migration is driven by nominal wages
differences and that all H earnings are remitted to the country of origin.16 Thus, migration
changes n and n* but not E and E*.
We start with the market clearing condition for a typical northern M-variety:
where R=w1-σµE/∆+φw1-σµE*/∆*, and ∆=nw1-σ+φn*w*1-σ, ∆*=φnw1-σ+n*w*1-σ.
Log differentiating this expression, yields (note all share variables such as sR, sE and sn lie
in the zero-one range):
where sR is share of a typical north firm’s total sales, R, that are made in the north and the
second expression follows due to the equal and opposite nature of all changes around
symmetry. Observe that at the symmetric outcome (i.e. sn=sH=½), sR exceeds ½ when
trade is not perfectly free, i.e. φ<1. Moreover, sR falls toward ½ as φ approaches unity;
specifically, sR=1/(1+φ) at sH=½.
By supposition, expenditure is repatriated so 0ˆ=
s, and given the definition of ∆:
where sM is the share of northern expenditure that falls on northern M-varieties. With
positive trade costs, sM exceeds ½ with the difference shrinking as φ increases; in fact
using the demand functions and symmetry we can show that 2(sM-½)=(1-φ)/(1+φ). Using
(1.14) in (1.13) with dsE=0 yields:
This is the “local competition” effect in isolation. Note that sR and sM lie in the zero-one
There are four salient points. First, since the denominator must be positive (since
4(sR-½)(sM-½) is always less than unity and σ>1) and the numerator must be negative,
northward migration always lowers the northern nominal wage and, by symmetry, raises
the southern wage. Second, this shows directly that migration is not, per se, destabilising.
When the demand or cost linkages are cut, as in this thought experiment, the symmetric
equilibrium is always stable despite migration. Third, the magnitude of this “competition
16 This may be thought of as corresponding to the case where H is physical capital whose owners are
immobile. Note also that under these suppositions, the model closely resembles the pre-economic
geography models with monopolistic competition and trade costs, e.g. Venables (1987) and Chapter 10 of
Helpman and Krugman (1989).
for consumers” effect diminishes roughly with the square of trade costs since as trade
free-ness rises, (sR-½) and (sM-½) fall. Specifically, 4(sR-½)(sM-½)=[(1-φ)/(1+φ)]2. Note
that in FKV terminology (sR-½) and (sM-½) are denoted as “Z” since at the symmetric
equilibrium both equal (1-φ)/(1+φ).
The final point is that in this thought experiment the break and sustain points are
identical; this can be seen by noting that sn doesn't enter (1.15). Both points equal φ=1
since the symmetric outcome is stable, and the core-periphery outcome is unstable for
any positive level of trade costs. When there are no trade costs, any locational outcome is
In the next thought experiment, suppose that, for some reason, H bases its
migration decision on nominal wages but spends all of its income in the region it is
employed. While this would not make much sense to a rational H-worker, the assumption
serves intuition by allowing us to restore the connection between production shifting
(dH=dn) and expenditure shifting dE without at the same time adding in the cost-linkage
(i.e. cost-of-living effect). Since E equals L+wH and this equals L+wn with our
normalisations, the restored term from (1.13) is:
The second expression follows since, from (1.8), w=1, n=½ and E=1/2µ at the symmetric
outcome. Using (1.16) and (1.14) in (1.13), we find:
Note that the denominator is always positive, since 0≤4(sR-½)(sM-½)≤1.
Six aspects of (1.17) are worth highlighting. First, the de-stabilising aspects of
demand-linked circular causality can be seen by the fact that the first term in the
numerator is positive. Second, the size of the de-stabilising demand linkage increases
with the M-sector expenditure share, µ. This makes sense since as µ rises, a given amount
of expenditure shifting has a bigger impact on the profitability of locating in the north.
Third, the size of this destabilising effect falls as trade gets freer since sR approaches ½ as
φ approaches unity. Fourth, the magnitude of the stabilising local-competition effect
erodes faster than the de-stabilising force since both sR and sM approach ½ as φ
approaches unity. Fifth, the symmetric outcome is stable with very high trade costs. To
see this observe that 4(sR-½)(sM-½)=2(sR-½)=1 at φ=0 and µ<1.
Finally, at some level of trade free-ness, namely φb’=(1-µ)/(1+µ), dw/dn is zero.
This critical value is useful in characterising the strength of agglomeration forces since it
defines the range of trade costs where agglomeration forces outweigh the dispersion
force. Thus an expansion of this range (i.e. a fall in the critical value) indicates that
agglomeration dominates over a wider range of trade costs.
The above thought experiments isolate the importance of the local competition
effect and demand-linked circular causality. The final force operating in the model works
through the cost-of-living effect. Since the price of imported varieties bears the trade
costs, consumers gain – other things equal – from local production of a variety. This
effect, which we dub the “location effect”, is a de-stabilising force. A northward
migration shock leads to production shifting that lowers the cost-of-living in the north
and thus tends to makes northward migration more attractive. To see this more directly,
we return to the full model with H basing its migration decisions on real wages and
spending their incomes locally. Log differentiating the northern real wage, we have
ˆˆ σµϖ−∆−= w. Using (1.14):
The second term is the cost-of-living effect, also known as cost-linked circular causality,
cost linkages, or backward linkages. Since this is positive, the cost-of-living linkage is
de-stabilising in the sense that it tends to make more positive the real wage change
stemming from a given migration shock. Moreover, consumers care more about local
production as µ/(σ-1) increase, so the magnitude of the cost-of-living effect increases as
µ rises and σ falls. Higher trade costs also amplify the size of the effect since sR rises
towards 1 as φ approaches zero.
Two observations are in order. Observe first that the cost-linkage can be separated
entirely from the demand and local competition effects. The first term in (1.18) captures
the demand-linkage and the local competition effect, while the second term captures the
cost-linkage. Second, note that the coefficient on
is positive – since 2(sR-½)≤1 – so the
net impact of the demand linkage and local competition effects on ω depends only on the
sign of (1.17).
The No-Black-Hole Condition
To explore stability at very high trade costs, we use (1.17) and set φ=0 to get
at sn=½ equals
. Stability requires this to be negative and
solving we see that this holds only when µ<(1-1/σ). If this, which FKV call the ‘no black
hole’ condition, holds, then the dispersed equilibrium is stable with very high trade costs.
Otherwise, the symmetric equilibrium is never stable.
1.4.3. The Break Point
We have seen that the magnitude of both the agglomeration and dispersion forces
diminish as trade cost fall, but the dispersion force diminishes faster than the
agglomeration forces. We also saw that when the no black hole condition holds, the
symmetric equilibrium is stable – i.e. the dispersion force is stronger than the
agglomeration forces – for very high trade costs.
Figure 1.3: Agglomeration and Dispersion Forces Erode with φφ
Figure 1.3 illustrates both of these facts. The bifurcation point (i.e. the level of
trade costs where the nature of the model’s stability changes) is where the agglomeration
and dispersion forces are equally strong.
Finally, noting that 2(sR-½)=2(sM-½)=(1-φ)/(1+φ), we can find the level of φ
where the bifurcation occurs by plugging (1.17) into (1.18), setting the result equal to
zero and solving for φ. The solution is:
The break point can be used as a metric for the relative strength of agglomeration forces.
For example, if a particular parameter change reduces φB, it must be that the change leads
the agglomeration forces to overpower the dispersion force at a higher level of trade
costs. This, in turn, implies that the change has strengthened the agglomeration forces
relative to the dispersion forces.
Note that from (1.19), the break point falls when µ rises and when σ falls. This
should make sense since µ magnifies both the demand-linked and the cost-linked
agglomeration forces, while a fall in the substitutability of varieties, i.e. a rise in 1/(σ-1),
magnifies the cost-of-living linked agglomeration (by strengthening the utility benefit of
local production). Of course, with free entry, 1/σ is also a measure of scale (see Appendix
1), so, loosely speaking, we can also say that an increase in equilibrium scale economies
magnifies the cost-of-living agglomeration force.
Magnitude of forces
Dispersion Forces (local competition effect)
(backward & forward linkages)
1.4.4. The Sustain Point
The sustain point is much easier to characterise since it involves the comparison
of levels rather than the signing of a derivative. Specifically, we evaluate w/P and w*/P*
at the CP outcome (we take sn=sH=1, although sn=sH=0 would do just as well) and look
for the level of φ where the two real wages are equal. Given our normalisation, w and P
equal unity at the CP outcome (to see this plug n=1 and n*=0 into (1.6) to find w=1 and
then use this and n=1 and n*=0 in the definition of P). Using the southern equivalent of
(1.6), we have (w*)σ=φµ(1+L)+µL/φ) at the CP outcome, where L is each region’s
endowment of the immobile factor and L=(1-µ)/2µ with our normalisations. Plainly, this
w* is a sort of ‘virtual’ nominal wage since no labour is actually employed in the south
when sH=1. Finally, in the south all M-varieties are imported when sH=1, so P*=φµ/(1-σ).
Putting these points together, the sustain point is implicitly defined by:
With some manipulation, this came be shown to be equivalent to the expression for the
sustain point in (1.12).
1.4.5. Comparing the Break and Sustain Points
The fact that the sustain point occurs at a higher level of trade free-ness than the
break point is well known and has been demonstrated in thousands of numerical
simulations by dozens of authors. Yet a valid proof of this critical feature of the model
was never undertaken.17
The most satisfying approach to proving that φS>φB would be direct algebraic
manipulation of expressions for the two critical points. This is not possible since φS can
only be defined implicitly as in (1.20). Instead, we pursue a two-step proof. First we
characterise the how the function, 1]2/)1()1[()( 2
/11 −−++= −
f, which is just a
transformation of the second expression in (1.20), changes with φ. This function is of
interest since φS is its root. With some work we can show three facts: that f(1)=0 and f’(1)
is positive, that f(0) is positive and f’(0) is negative, and that f(.) has a unique minimum.
Taken together, this means f has a unique root between zero and unity. In short, it looks
like the f drawn in Figure 1.4. Next, we show that f(φB)<0, which is only possible if
φS>φB, given the shape of f(.). To this end, observe that f(φB) is a function of µ and σ.
Call this new function g(µ,σ) and note that the partial of g with respect to µ is negative
and g(0,σ) is zero regardless of σ. The point of all this is that the upper bound of g, and
thus the upper bound of f(φB), is zero. We know, therefore, that for permissible values of
µ and σ, φS>φB.
17 The first draft of the excellent paper by Peter Neary, Neary (2000), was seen by us before we wrote this
chapter. That draft contained a brief proof in a footnote that turned out to be incorrect. Frederic Robert-
Nicoud showed the proof’s error and provided a correct proof, which Peter Neary incorporated (with
accreditation) in subsequent drafts of his paper.
Figure 1.4: Proving the φφB<φφS.
1.5.1. When Does Symmetry Break? Pareto Dominance and Migration
The analysis to this point has focused only on local stability, as is true of the vast
majority of the literature. This is not enough. For instance, when does symmetry would
break as trade costs fell gradually from prohibitive to negligible? The standard answer is
that it breaks at the break point. This is not necessarily true. For levels of trade free-ness
between φS and φB, there are three locally stable equilibrium: sn=1/2, sn=1 and sn=0. In
game theory, where multiple equilibria is viewed as a problem, it is common to apply the
‘Pareto refinement’. That is, if a particular equilibrium is Pareto dominated by another,
there is some presumption that agents would be able to co-ordinate sufficiently to avoid
the inferior outcome. The technical name of the equilibria that survive this refinement are
“coalition proof equilibria” (Bernheim, Peleg and Whinston, 19??). As it turns out, sn=1/2
is not coalition proof when φS<φ<φB.
With φ between φS and φB, all workers are better off at either CP outcome than
they are at the symmetric outcome – due to the cost-of-living effect. One might therefore
presume that a sufficiently large coalition of workers would agree to migration once trade
costs got low enough to make the CP outcome locally stable. This would be rational since
if they all did move their instantaneous real wage would rise. All this goes to challenge
the standard analysis that claims that starting with very high trade costs, the economy
remains at the symmetric outcome until the break point is reached. If coalitions of
workers can migrate, it is possible – and indeed would be very rational – for symmetry to
00.2 0.4 0.6 0.8 1
collapse when trade costs fall to the sustain point. More formally this just says that while
symmetry is locally stable when φS<φ<φB, it is not globally stable.
This brings us to issues of global stability.
1.5.2. A Caveat on Full Agglomeration
Only one dispersion force operates in the CP model and this (local competition)
becomes very weak as trade gets very free. As a result, the model predicts that
sufficiently high levels of trade free-ness are inevitably associated with full
agglomeration. The world, however, is full of dispersion forces – comparative advantage,
congestion externalities, natural resources, “real” geography such as rivers, natural ports,
etc – and these can change everything.
The point is that, as noted above, the model’s agglomeration forces also decrease
with trade costs. This implies that for low enough trade costs other dispersion forces that
are not eroded by trade free-ness, such as comparative advantages, will dominate the
location decisions of firms with trade becomes sufficiently free. In the literature this is
called the U-shaped result. Dispersion is the likely outcome both with trade costs are very
high and when they are very low. This appealling feature is not, unfortunately, present in
the simple CP model.18
What all this means is that the core-periphery model should not be construed as
predicting that the world must end up in an agglomerated equilibrium as trade costs are
lowered. Rather the model predicts that dramatic changes in location may happen for
some levels trade costs
1.6. Global Stability Analysis: Liaponov’s Direct Method
Local stability analysis is fine for most uses, but it is not sufficient for fully
characterising the model’s behaviour when sH is away from a long-run equilibrium (e.g.
when the process of agglomeration is ‘en route’). The economic geography literature
typically avoids discussing what happens between long-run equilibria, but where is does
it relies on a heuristic approach. Namely, it is asserted that the system approaches the
nearest stable equilibrium that does not require crossing an unstable equilibrium. We
show here that this heuristic approach can be justified formally.
At a high level of abstraction, the CP model is just a non-linear differential
equation with sH, or sn (since sH= sn) as the state variable. One simple approach to global
stability analysis of non-linear differential equations is called Liaponov’s direct method.
Instead of working with a potentially complicated function of the state variable (the
solution to the non-linear equation for sn, in this case), one works with a simple function
– defined on a specific region – that attains its minimum at the long-run level of sn. If the
simple function (called the Liaponov function) and its domain are chosen judiciously, one
can show that the value of the Liaponov function continuously approaches its minimum
as time passes and that this implies that the state variable also approaches its long-run
18 See FKV for various modificaions that lead to the U-shaped result.
equilibrium as time passes. This is sufficient for showing that the system is globally
stable in the region (see Beavis and Dobbs, 1990 p.167 for details).
The dynamics of the CP model depends upon the level of trade free-ness and there
are three qualitative cases (see Figure 1.2). When φ is very low, only the symmetric
equilibrium is stable; when φ is very high, only the CP equilibria are stable. The most
interesting case, from a global stability point of view, is the case of intermediate trade
costs. For an intermediate level of trade free-ness, the model has five equilibria, three of
which are stable (the symmetric and the two CP outcomes) and two of which are
unstable. This case is shown in Figure 1.5, where the unstable equilibria are labelled U1
What we wish to show is that, even in the five-steady-states case, the system is
globally stable in the sense that the system always converges to one steady state or
another regardless of initial conditions. We also want to show that the system approaches
the nearest stable steady-state that does not require crossing of an unstable steady state.
Figure 1.5: The “Wiggle Diagram”
Consider first stability in the open set sn∈(U1,U2). The Liaponov function we
choose is (sn-1/2)2/2. This satisfies the regularity conditions of Theorem 5.24 in Beavis
and Dobbs (1990), namely the equilibrium point and initial point are in the set, the
function is always positive on the set and the value of the function is zero at the
equilibrium. Most importantly, 0)2/1(<−= nn ssV&
& for all t and for all non-equilibrium
values of sn in the set. To see this, note that sn is increasing when sn is less than ½, but
decreasing when sn exceeds ½. Since V is always decreasing and attains its minimum at
the symmetric steady state, we know that sn converges to the symmetric steady state
whenever the initial value is in the sn∈(U1,U2) range. This range is sometimes called the
symmetric equilibrium’s ‘basin of attraction’.
Next consider stability in the sn∈(U2,1] interval with (sn-1)2/2 as the Liaponov
function. This function meets all the regularity conditions and time-derivative condition,
so we know that sn∈(U2,1] is the basin of attraction for the core-in-the-north CP outcome.
Similar reasoning implies that sn ∈[0,U1) is the basin of attraction for the core-in-the-
south CP outcome.
Finally, analogous reasoning can show that the CP model is globally stable in the
two simpler cases when only the symmetry outcome is stable and when only the CP
outcome is stable. Moreover, in the latter case, it is straightforward to establish that
(1/2,1] and [0,1/2) are, respectively, the basins of attractions for the core-in-the-north and
core-in-the-south CP outcomes. See Baldwin (2000) for details.
The logic of demand-linked agglomeration depends crucially upon market size, so
it is natural to wonder whether the key results – catastrophic agglomeration and
locational hysteresis – would hold when regions are intrinsically asymmetric in terms of
size. Another type of asymmetry to be considered is that of trade free-ness. That is, if one
nation’s φ is large but both fall, do we still observe catastrophies and overlaps?
1.7.1. Size Asymmetries
A nation’s economic size depends on how much L and H it has. Since H is mobile
and its international division is endogenous, intrinsic size asymmetries must come from
different endowments of the immobile L. To this end, we assume that the two regions are
endowed with different stock of L and to be concrete, the South is ‘bigger’ in the sense
that L*=L+ε with ε>0.
Intuition the how size-asymmetry matters can be had by considering a small
change to a situation that starts out fully symmetric in terms of the division of both H and
L. Formally, this involves consideration of a small perturbation, dε, of the fully
symmetric equilibrium where initially ε=0 and sn=1/2. Mechanically, the ε enters into the
equilibrium conditions via the definition of E’s, namely E=L+wH and E*=L*+ε+w*H*.
Since the E’s enter the excess supply equations (via the demand functions) and the excess
supply equations determine the nominal wage, a change in ε affects w and w*. To
quantify this, we totally differentiate the two excess supply equations with respect to w,
w* and ε and evaluate the result at sn=1/2 and ε=0. Solving these yields expressions for
dw/dε and dw*/dε and these tell us how equilibrium nominal wages are affected by a
slight size asymmetry. Next we totally differentiate the real wage gap, Ω, with respect to
the nominal wages and plug in the expressions for dw/dε and dw*/dε. The result is an
expression that tells us how the real wage gap at full symmetry would be affected by a
slight size asymmetry:
Given the standard restrictions on the parameters (σ>1 and 0<µ<1), (1.21) is
negative by inspection. Since the real wage gap is zero at the initial point of full
symmetry, and Ω=0 is a long-run equilibrium condition, we see that even a slight size
asymmetry rules out the possibility of an even division of industry. In particular, if sn
were ½, the southern real wage would be slightly higher so north-to-south migration
would occur. What is the new equilibrium division on H? Unfortunately, the intense
intractability of the CP model means that numerical simulation of the model for specific
values of µ, σ and ε is the only way forward.
Figure 1.6 plots the real wage gap, Ω, against sH (the share of mobile workers in
the north) for various levels of trade free-ness taking ε=.01, µ=.3 and σ=5. When φ is
very low, say, 0.1, or very high, 0.9, we have three long-run equilibria. The two core-
periphery outcomes, sH=1 and sH=0 – which are always equilibria given the migration
equation (1.2) – and an interior equilibrium at the point where the plot of Ω crosses the x-
axis. As shown for φ=0.1 (this corresponds to trade costs of almost 80%), Ω is steeply
declining in sH over the whole range of sH. This tells us that only the interior equilibrium
is stable since Ω is positive at sH=0 and negative at sH=1 (see (1.9) for a formal statement
of local stability criteria). When trade is very free, say φ=0.9 (i.e. 3% trade costs), we also
have a unique interior equilibrium, but Ω is steeply rising, so only the two CP outcomes,
sH=1 and sH=0, are stable and the interior equilibrium is unstable.
Figure 1.6: Wiggle Diagram with Size Asymmetry.
For intermediate values of φ we have outcomes with one, two or three interior
equilibria. For example, when φ=0.212 there are two interior equilibria marked A and B
in the diagram; the first is unstable while the second is stable. For φ=0.23, we have three,
C, D and F of which only the middle one is stable. And for φ=0.24, the only one interior
equilibrium, point E, is unstable. Plainly then, the asymmetric-size case presents a richer
array of outcomes than does the symmetric case.
These simulation results can be parsimoniously illustrated in a diagram similar to
the Tomahawk diagram. Figure 1.7 plots the long-run equilibrium division of industry on
the vertical axis for all possible levels of trade free-ness. Interestingly, we see that size
asymmetry “breaks the handle” of the tomahawk from Figure 1.2 into two pieces and
rotates the pieces in opposite directions. More precisely, from the above equation, we see
that dΩ/dε=0 at two values of trade free-ness, φ=-(1-µ)/(1+µ) and φ=1. The first of these,
while outside the range of economically meaningful φ, tells us that the fulcrum for the
rotation of the right-hand part is -(1-µ)/(1+µ); the left-hand part rotates around φ=1.
Figure 1.7: Breaking the Tomahawk with Size Asymmetry.
Notice that we now have two sustain points and a single break point and the stable
interior equilibrium is no longer a straight-line as in Figure 1.2. These features
significantly enrich the range of possibilities compared to the symmetric CP model. For
instance, suppose we tell the usual story of how falling trade costs can affect the location
of industry. Starting with very high trade costs and only slight size asymmetries,
progressive reductions in trade costs have only a slight location impact, with some
industry moving from the small region to the large region. However as the level of trade
free-ness approaches the break point, φB, the location effect of a marginal increase in
trade free-ness is greatly magnified with a large share of northern industry de-locating to
the big region (the south). Once φB is surpassed, industry either all moves to the north or
all to the south. Unlike in the symmetric case, the full agglomeration in the big region is
much more likely. In short this model displays the catastrophic features of the CP model,
but also display a richer, pre-catastrophe behaviour.
The hysteresis features of this model are also richer. In the symmetry CP model,
there is a single sustain point, so both regions become able to sustain full agglomeration
at the same time. With size-asymmetry, by contrast, the big region is able to sustain the
core at a higher level of trade cost than is the small region. What this means is that a
some intermediate level of trade costs, a sufficiently large location shock could switch the
outcome from a fairly even division of industry to one dominated by the big region, but
no shock could shift the outcome to having the core in the small region. At a somewhat
higher level of trade free-ness, however, a big location shock could – as in the symmetric
CP model – shift industry to either extreme.
Further numerical simulation (not reported) shows that when the size asymmetry
gets larger, the ordering of the break and sustain points can change. Specifically, the large
region’s sustain point always comes at a lower φ than the small region’s but with very
asymmetric regions the break point is between the two sustain points.
1.7.2. Asymmetric Trade Costs
A second type of asymmetric involves trade costs. As it turns out, the qualitative
results for this type of asymmetric are quite similar to those describe above, so we cover
trade-cost-asymmetry rather quickly.
Assume the two regions differ in terms of their openness to imports (i.e. φ*=φ-δ
δ>0) with the North more open than the South. The first step to understand what happens
is again to consider a small perturbation dδ of the symmetric equilibrium sn=1/2 which
where negativity is granted by the no-black-hole condition.
This expression implies that a small decrease in southern openness makes the
North-South real wage gap negative. Since at sn=1/2 Ω was zero, the perturbation triggers
migration of industrial workers from North to South. As before, this asymmetry
eliminates the sn=½ equilibrium and creates a situation with two sustain points and a
1.8. Forward-Looking Expectations
Perhaps the least attractive of the CP-model assumptions concerns migrant
behaviour. Migrants are assumed to ignore the future, basing their migration choices on
current real wage differences alone. This is awkward since migration is the key to
agglomeration, workers are infinitely lived, and migration alters wages in a predictable
manner. While the shortcomings of myopia were abundantly clear to the model’s
progenitors, the assumption was thought necessary for tractability. This turns out not to
CP-model dynamics are intractable for two distinct reasons. Forward-looking
expectations forces consideration of the very difficult issues of global stability in non-
linear dynamic systems with multiple equilibria. Although these are difficult, they are
also important and interesting, as Matsuyama (1991) – the first to consider them formally
– shows. For instance, such considerations open up a very important set of possibilities
such as self-fulfilling expectations (i.e. spatial reallocations that are unrelated to changes
in the economic environment but rather are triggered by shocks to expectations) and the
idea that policies can work by deleting equilibria. The second source of intractability is
model-specific; the CP model cannot be reduced to a set of explicit differential equations
since ω-ω* cannot be written as an explicit function of the state variable, sn.19 This, in
turn, is because (1.6) cannot be solved analytically.
This section considers a combination of analytic and numerical techniques that
allow us to see how the standard CP model is changed when migrants are allowed to
think about the future .
1.8.1. The Dynamic Problem and Optimal Migration Behaviour
The first task is to find a way of nesting the standard CP model within a closely
related model that allows forward-looking expectations. To this end, none of the CP
model assumptions, except the assumption concerning migrant myopia and the definition
of households (i.e. the representative consumers), are altered. In particular, assume there
are N households that are identical and endowed with Hw/N units of industrial labour; H
and L/N units of agricultural labour. Hi units of the household’s H are employed in the
north and 1/N-Hi are employed in the south (recall that Hw is normalised to unity). A
representative household’s intertemporal preferences are defined by:
where ρ is the subjective discount rate, and U is as in (1.1).
Migration is assumed to be costly, specifically the migration cost is quadratic in
the flow of south-to-north migration, namely ii Hm &
≡(this is a standard assumption
intended to reflect congestion costs). Migration costs are also related to the existing inter-
regional distribution of H. Thus, migration costs are (
mi2/2) times (1/Hi)(1/[1/N-Hi]),
measures overall migration costs. Given this, optimal migration behaviour is
simple to derive.20
The representative household divides its labour between north and south to
maximise its real earnings net of migration cost. Observing that the real wage is an index
for worker’s instantaneous utility (i.e. P is a perfect price index and w is the only source
of income for the mobile factor), the representative household chooses migration to
where the first term in the large brackets is the representative household’s income from
its immobile factor. This allows for almost any sort of migration behaviour, including the
19 Since H=n with our normalisations, either sH or sn can be taken as the state variable. Since other
economic geography models can be written with sn as the state variable, we prefer sn to bolster
comparability across models.
20 For justification of these assumptions see Baldwin (2000).
possibility that migrants will return and re-migrate.21 Ignoring constants, the current
valued Hamiltonian for the problem is Hiω+(1/N-Hi)ω*-
where W is the co-state variable that captures the asset value of migration. The solution
to this is characterised by three necessary conditions )/1(iii HNWHm−= and
&, which must hold at all moments, and an endpoint condition
Using our normalisations and household symmetry, we have H=sH=sn=NHi and
mss Hn==&& . Using this in the first necessary condition, and absorbing N terms into γ=
the aggregate migration equation becomes:
As usual, W is governed by an asset-pricing-like expression:
1.8.2. Myopic and Forward Looking Expectations
If migrants have rational, forward-looking expectations, (1.25) and (1.26)
characterise their optimal behaviour. If migrants are myopic, i.e. have static expectations,
they assume that the current real wage gap will persist forever, and (1.26) can the be
solved to yield W=(ω-ω*)/ρ. Using this in (1.25) implies:
(1.27)nnnn ssss )(/)1(*)(ωωγρωω −=−−=
The second expression – which is identical to (1.2) given our normalisations –
follows from the first expression by choice of time units (such that ργ=1). This result
shows myopic behaviour can be nested in the pair differential equations, (1.25) and
(1.26), that characterise optimal migration behaviour with forward-looking behaviour. In
particular, myopic behaviour is tantamount to static expectations. For details, see
1.8.3. Stability Analysis with Forward-Looking Workers
We turn next to the local and global stability properties of the CP model with
Local stability is assessed by using a linear approximation to the non-linear
system given by (1.25) and (1.26). The linearised system is )( ss
xxJx −=& where x≡(sn,W)T
21 The main restriction is that we rule out an infinite number of migrations in a finite period. Moreover
since the co-state variable must be a continuous function of time, the migrants cannot expect to change
their migration time path in the future.
and J is the Jacobian matrix (i.e. matrix of own and cross partials) evaluated at a
particular steady state. Specifically, J is:
Local stability is determined by checking J’s eigenvalues at the symmetric and CP
outcomes. As usual, saddle path stability requires one negative eigenvalue and one
positive eigenvalue. If the eigenvalues are complex, then the test involves the signs of the
One useful fact reduces the work. A standard result in matrix algebra is that the
determinant of J equals the product of the eigenvalues (Beavis and Dobbs, 1990 p.161).
Thus the system is saddle-path stable, if and only if det(J)<0. The determinant det(J) is
equal to (dΩ/dsn)sL(1-sn)/γ-ρW(1-2sn)/γ, so for the symmetric equilibrium – where 1-2sn
is zero – the stability test is (dΩ/dsn)/4γ<0 and in the CP outcome – where sn(1-sn) is zero
– the stability test is ρW/γ<0; in each case the expressions and derivatives are evaluated
at the appropriate steady state. Noting that W in the CP equilibrium equals ωCP-ωCP */ρ,
this shows a striking result. Informal local stability test for the CP model with static
expectations – viz. (1.9) – is exactly equivalent to the formal local stability test for the CP
model with forward-looking expectations.
An important and somewhat unexpected corollary of this result is that the break
and sustain points are exactly the same with static and with forward-looking expectations.
When trade costs are such that the CP model has a unique stable equilibrium,
local stability analysis is sufficient. After any shock, W jumps to put the system on the
saddle path leading to the unique stable equilibrium (if it did not, the system would
diverge and thereby violate a necessary condition for intertemporal optimisation). For φ’s
where the model has multiple stable steady states things are more complex. With multiple
stable equilibria, there will be multiple saddle paths. In principle, multiple saddle paths
may correspond to a given initial condition, thus creating what Matsuyama (1991) calls
an indeterminacy of the equilibrium path. In other words, it is not clear which path the
system will jump to, so the interesting possibility of self-fulfilling prophecies and sudden
takeoffs may arise. These possibilities are explored next.
Recent advances in computing speed and simulation software have made it
possible to numerically characterise non-linear systems with multiple steady states to a
very high degree of accuracy. There are two key tricks to doing this; it is much easier to
find the unstable saddle path than the stable saddle path, and the stable path becomes the
22 See the appendix to Barro and Sala-i-Martin (1995) for details and an excellent exposition of local
stability and phase diagram analysis.
unstable path in reverse time.23 Numerical techniques are also used to solve the second
source of intractability, namely the fact that wω-ω* cannot be written as an explicit
function of the state variable. To get around this, the computer is used to solve the model
for the exact values of Ω≡ω-ω* corresponding to a grid of values of sn∈[0,..1]. A very
high order polynomial of sn is then fitted to these actual values. The result is an explicit
polynomial function, Ω[sn], in the simulations that follow, a 17th order polynomial was
fitted to 25 values of Ω.24
Numerical simulation (in reverse time) enables us to find the saddle paths for
various parameter values; we always assume φS<φ<φB so that the system is marked by
three stable steady states. Three qualitatively different cases are consider for the
migration cost parameter γ . In all simulations we take σ=5, µ=4/10, ρ=1/10 and φ=1/10.
The first case is when γ, the migration cost parameter, is very large, so horizontal
movement is very slow. This is shown in Figure 1.8. Importantly, there is no overlap of
saddle paths in this case, so the global stability analysis with static expectations is exactly
right. That is, the basins of attraction for the various equilibria are the same with static
and forward-looking expectations. This is an important result. It says that if migration
costs are sufficiently high, the global as well as the local stability properties of the CP
model with forward-looking expectations are qualitatively identical those of the model
with myopic migrants.
The second case, shown in Figure 1.9, is for an intermediate value of migration
costs. Here the saddle paths overlap somewhat since the Jacobian evaluated at either
unstable equilibrium has complex eigenvalues – this means that the system spirals out
from U1 and U2 in normal time. (The figure shows only the saddle paths in the right side
of the diagram since the left side is the mirror image of the right).
The existence of overlapping saddle paths changes things dramatically, as
Krugman (1991c) showed. If the economy finds it itself with a level of sn that lays in the
overlap, namely the interval (A,B) shown in the figure, then a fundamental indeterminacy
exists. Both saddle paths provide perfectly rational adjustment tracks. Forward-looking
workers who are fully aware of how the economy works could adopt the path leading to
the symmetric outcome. It would, however, be equally rational for them to jump on the
track that will take them to the CPN outcome.
Figure 1.8: Global stability with forward-looking expectations and high migration
23 Dynamic systems marked by saddle path stability always have unstable saddle paths. In linear systems
the former correspond to the positive eigenvector, the latter to the negative eigenvector. See Baldwin
(2000) for details.
24 Algorithms showing how to find saddles paths and approximate Ω[sn] are available from the web site
Figure 1.9: Global stability with forward-looking expectations and intermediate
Which track is taken cannot be decided in this model. Workers individually choose a
migration strategy taking as given their beliefs about the aggregate path. Consistency
requires that beliefs are rational on any equilibrium path. That is, the aggregate path that
results from each worker’s choice is the one that each of them believes to be the
equilibrium path. Putting it more colloquially, workers chose the path that they think
other workers will take. In other words, expectations, rather than history, can matter.
Because expectations can change suddenly, even with no change in environmental
parameters, the system is subject to sudden and seemingly unpredictable takeoffs and/or
reversals. Moreover, the government may influence the state of the economy by
announcing a policy, say a tax, that deletes an equilibrium even when the current state of
the economy is distant to the deleted equilibrium.
While it is difficult to fully characterise the constellation of parameters which
corresponds to the overlap, it is easy to find a sufficient condition for there to be some
overlap of saddle paths. If the eigenvalues of the Jacobian evaluated at the unstable
equilibria are complex, then there must be some overlap. The eigenvalues at U2 are
2γρρ LLL ssdsd−Ω−± , so we get complex roots when migration costs are
sufficiently low, namely when:
In words, the possibility of history-versus-expectations arises when the costs of migration
are low relative to the patience of workers (i.e. 1/ρ2) and the impact that migration has on
the real wage gap (i.e. dΩ/dsn) is large.
The final case, Figure 1.10, is the most spectacular. Here migration costs are very
low, so horizontal movement is quite fast. As a result, the saddle path for CPN originates
from U1 rather than U2. Interestingly, the overlap of saddle paths includes the symmetric
equilibrium. This raises the possibility that the economy could jump from the symmetric
equilibrium onto a path that leads it to a CP outcome merely because all the workers
expected that everyone else was going to migrate. Plainly, this raises the possibility of a
big-push drive by a government having some very dramatic effects.
Finally, note that the region of overlapping saddle paths will never include a CP
outcome. Thus, although one may ‘talk the economy’ out of a symmetric equilibrium,
one can never do the same for an economy that is already agglomerated.
Figure 1.10: Global stability with forward-looking expectations and intermediate
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APPENDIX 1: ALL YOU WANTED TO KNOW
ABOUT DIXIT-STIGLITZ BUT WERE AFRAID TO
The CES preferences at the heart of the Dixit-Stiglitz monopolistic competition
model can be expressed in terms of discrete varieties or a continuum of varieties. To wit:
1> cCc CCUN
where N is the mass of varieties in the former (continuum version) and the number of
varieties in the latter (discrete version). The corresponding indirect utility functions are
both E/PM where the price indices are, respectively:
1> pPp P
PM is called a ‘perfect’ price index since it translates nominal expenditure, E, into utility.
The first order condition of utility maximisation yields (with discrete varieties):
Multiplying both sides by cj, summing across varieties and using the budget constraint
E= Σipici yields one expression for the Lagrangian multiplier λ=CM/E. Alternatively,
isolating cj on the left-hand side, multiply both sides by pj, sum across varieties and use
the budget constraint, we get λ=(Σipi1-σ)1/σ(E-1/σ)CX/(Σici1-1/σ). Plugging the first or second
λ expression into the first-order condition yields the inverse or direct demand curves,
respectively. These are:
j∀== ∑∑ ==
It is convenient to use the indirect demand function when assuming Cournot competition
and to use the direct demand function when working with Bertrand competition.
The CES utility function is often referred to as “love of variety” preferences. To
understand why, we show that the same level of expenditure spread over more varieties
increases utility. If all varieties are priced at ‘p’, consumption of a typical variety is
E/(Np). Substituting this into the utility function implies that U=N1/(σ-1)(E/p). Utility rises
with N, so in this sense, consumers love variety for variety’s sake. Moreover, even if
each variety is priced differently, adding a new one increases utility if prices of the
existing varieties are unchanged; this is very easily seen by using the expression for the
perfect price index in the discrete case.
A1.1 Dixit-Stiglitz Competition, Mill Pricing and Firms’ First Order
Dixit-Stiglitz monopolistic competition is highly tractable since a firm’s optimal
price is a constant mark-up over marginal cost. This is unusual since in most forms of
imperfect competition, the optimal price-marginal cost mark-up depends upon the degree
of competition, with the mark-up increasing as the degree of competition falls. For
example, the optimal mark-up often depends upon the firm’s market share, but since the
market share depends upon prices, one needs to solve all firms’ first order conditions
simultaneously. If additionally the number of firms is determined by free entry, finding
equilibrium prices can require the simultaneous solution of many equations, some which
will be non-linear. Fixed mark-ups permits us to avoid all this.
To get started, consider Cournot competition among N firms in a single market,
with each firm producing a symmetric variety subject to a homothetic cost function. The
typical firm’s objective function is revenue, pjcj, minus costs, (amcj-F)w, where F is the
fixed cost, am is the variable cost and w is the wage. Using the discrete-varieties version
of the indirect demand function, the Cournot first order condition is:
where “s” is the market share of the typical variety; with symmetry s=1/N. Using the
direct demand function and Bertrand conjectures yields a similar condition as shown.
Under both conjectures, the perceived elasticity, ε, falls as ‘s’ rises.
Note that as long as ‘s’ is not zero, the degree of competition does affect the
mark-up and thus pricing behaviour. For example, with symmetry and Cournot, the
equilibrium mark-up is (1-1/σ)(1-1/N), so the equilibrium mark-up falls as the number of
competitors rises. This is called ‘small-group’ monopolistic competition. An interesting
extreme case – which is at the root of the Dixit-Stiglitz monopolistic competition – is
where N rises to infinity and the perceived elasticity ε equals σ (under both Cournot and
Bertrand). Since the perceived elasticity is invariant to N, the mark-up is constant. This
extreme case is what Chamberlain called the “large-group” case and it is what Dixit-
Stiglitz monopolistic competition assumes. Four comments are in order.
First, note that with an infinite number of atomistic competitors – i.e. under Dixit-
Stiglitz assumptions – equilibrium pricing does not depend upon the typical firm’s
conjecture about other firms’ reactions. Bertrand and Cournot conjectures produce the
same result. While this is convenient, it is a strong assumption that rules out many
interesting effects, such as the pro-competitive effect.
Second, in the discrete varieties version of Dixit-Stiglitz preferences (assumed in
the original 1978 article), one must assume that N is large enough to approximate ε with
σ. With the continuum of varieties case, there are an uncountable infinity of varieties, so
“s” is automatically zero.
Third, the invariance of the Dixit-Stiglitz mark-up to changes in the number
(mass) of firms is easily understood. One starts with the assumption that the number of
competitors is infinite, so adding in more competitors has no effect. Infinity, after all, is a
concept, not a number.
Fourth, the invariance of the mark-up leads to so-called mill pricing. That is, if it
costs T1 to ship the goods to market 1, and T2 to ship them to market 2, firms will fully
pass the shipping costs on to consumer prices, so the ratio of consumer prices in market 1
to market 2 will be T1/T2. This is called mill pricing, or factory gate pricing, since it is as
if the firm charged the same price “at the mill” or at the factory gate, with all shipping
charges being born by consumers. Another way of saying this is that with mill pricing, a
firm’s producer price is the same for sales to all markets.
A1-2: Operating Profits, Free Entry and the Invariance of Firm Scale
One extremely handy, but not very realistic, aspects of Dixit-Stiglitz monopolistic
competition is the invariance of equilibrium firm scale. This is a direct and inevitable
implication of the constant mark-up and free entry.
Plainly, a fixed mark-up of price over marginal cost implies a fixed operating
profit margin. It is not surprising, therefore, that there is a unique level of sales that
allows the typical firm to just break even, i.e. to earn a level of operating profit sufficient
to cover fixed costs. The first order condition (the pricing equation) can be arranged as:
The second to last expression shows that operating profit, (p-wax)c, equals an invariant
profit margin (namely, 1/σ) times the value of consumption at consumer prices. The last
equation is derived using the formula for the equilibrium price. The constancy of
equilibrium firm scale, i.e. the volume of sales/production necessary for a typical firm to
break even is obvious when the scale economies take the familiar form of a linear cost
function, namely w(F+axc), where wF is the fixed cost and waxc is the variable cost. The
zero profit condition in this case is just:
Observe that equilibrium firm scale depends on two cost parameters, F and ax, and a
demand parameter, σ.
As it turns out, the invariance of equilibrium scale economies demonstrated above
is quite a general proposition, at least for one common measure of scale, viz. the scale
elasticity. As long as the price-marginal cost mark-up is fixed and the zero profit
condition holds, the scale elasticity, i.e. χ≡(dC/dx)(x/C), where x is firm output/sales and
C is the cost function, must be constant. To see this, note that with zero profit, price must
equal average cost, so the first order condition can be written as AC/MC=(1-1/σ), where
MC and AC are marginal and average cost respectively. But, (dC/dx)(x/C) is just
MC/AC, so 1/χ=(1-1/σ).
Note that, the scale elasticity is a measure that has its limitations. For instance, if
the cost function in not homothetic in factor prices, a given scale elasticity does not
coincide one-to-one with firm size. For instance, if capital is used only in the fixed cost
and labour only in the variable costs, then the scale elasticity is (rF/(waxx)+1)-1. Even if
this is constant at, say, 1-1/σ, the firm size that corresponds to this depends upon the r/w
ratio. Since trade costs can in general affect factor prices, this means that trade costs can
also affect firm scale, even in the Dixit-Stiglitz model.
A1-3: Invariance of Firm Scale with Trade
The simplicity that comes with Dixit-Stiglitz monopolistic competition is
especially apparent with dealing with multiple markets. In particular, when we assume
that trade costs are “iceberg” in nature (i.e. are proportional to marginal production costs
since a fraction of shipped goods disappear in transit), solving the multi-market problem
is no more difficult than solve the single market problem.
To understand the source of the simplifications, we start with a more general set
of assumptions. Suppose there are two markets, local and export, and that it costs T* to
ship one unit of the good to the export market and T to ship it to the local market. These
costs are not of the iceberg type.
A typical firm has p(1-1/σ)=(wax+T) and p*(1-1/σ)=(wax+T*) as its first order
conditions, where p* is the consumer price in the distant market. Rearranging these
conditions shows that operating profit – which we denote at π -- is proportional to the
value of retail sales, R. Specifically, π=R/σ, so the free entry condition requires that
R/σ=wF as in the single market case without trade costs. However, now R=pc+p*c*,
where c and c* are consumption in the local and export markets. Rearranging, we have
that c+c*=wFσ/p-ψc*, where 1+ψ≡p*/p and from the first order conditions, p*/p equals
(wax+T*)/(wax+T). The left hand side is clearly not constant because the right-hand side
is not. Indeed, in general both terms on the right-hand side may vary.
What is needed to make c+c* invariant to trade costs? If the cost function is
homogenous, the fixed costs wF is proportional to the price (recall that price is
proportional to marginal cost), so wFσ/p will not vary with relative factor prices.
Nevertheless, scale will vary since both ψ and c* vary with trade costs. To make ψc*
constant, we assume that trade costs are “iceberg” in the sense that a certain fraction of
each shipment disappears in transit. This makes trade costs proportional to marginal cost.
For example, if marginal costs are wax(1+t*) for export sales and wax(1+t) for local sales,
then ψ equals (t*-t)/(1+t). In this case, we can without further loss of generality absorb
1+t into the definition of ax and define trade costs as zero for local sales and t”=t*-t for
distant sales (this is standard practice). Moreover, with iceberg costs we have that
(1+t”)c* equals the quantity produced for the distant market since the quantity produced
and shipped is always 1+t” times consumption.
In summary, the invariance of firm size (as measured by production) to trade costs
is a result that is very sensitive to special assumptions and functional forms. Trade costs
must be iceberg, the cost function must be homothetic and equilibrium prices must be
proportional to marginal costs (this in turn requires mill pricing).