The logic of demand- linked agglomeration depends crucially upon market size, so it is natural to wonder whether the crucial 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 larger but both φ’s fall, do we still observe catastrophes?
C.1 Asymmetric Sizes
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 endowme nts of the immobile L. To this end, we assume that the two regions are endowed with different stocks of L and to be concrete, the south is
‘bigger’ in the sense that L*=L+ε with ε>0.
Intuition on 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 market-clearing conditions (via the demand functions) and the market-clearing conditions determine nominal wages, a change in ε will generally affect w and w*. To quantify this, we totally differentiate the two market-clearing conditions 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, i.e. Ω=ω-ω*, at full symmetry would be affected by a slight size asymmetry. The result is:
(A-11) 0
Given the standard restrictions on the parameters (the no black hole condition and σ>1 and 0<µ<1), (A-11) 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 2.10 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 (2-3) – 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 (2-18) 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.
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.
Figure 2.10: Wiggle Diagram with Size Asymmetry.
These simulation results can be parsimoniously illustrated in a diagram similar to the Tomahawk diagram, namely Figure 2.11. This 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 2.4 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.
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 the symmetric-size case.
These features significantly enrich the range of possibilities compared to the
0.02
-0.02 0
φ=0.1
φ=0.9 φ=0.24
φ=0.23
s
HΩ
Ω
φ=0.212
A 1
B
C D E
F
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 relocating 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.
Figure 2.11: The Broken Tomahawk: Size Asymmetry in the CP Model
The hysteresis features of this model are also richer. In the symmetric 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 at 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.
C.2 Asymmetric Trade Costs
A second type of asymmetry involves trade costs. As it turns out, the qualitative results for this type of asymmetry are quite similar to those described above, so we cover trade-cost-asymmetry rather quickly.
φ sH
1 1
0 1/2
φsN φsS φB
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. Calculations similar to those described for the size-asymmetry case yield:
] 0
where negativity is guaranteed 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 break point.