B.2 “Proving” the Tomahawk Diagram
B.4 Forward-Looking Expectations
2.5.7 Stability Analysis with Forward-Looking Workers
We turn next to the local and global stability properties of the CP model with forward-looking migrants.
Local Stability
Local stability is assessed by using a linear approximation to the non-linear system given by (A-9) and (A-10). The linearised system is x&=J(x−xss) where x≡(sn,W)T 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 real parts.29
One useful fact reduces the work. A standard matrix algebra result 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
29 See the appendix to Barro and Sala -i-Martin (1995) for details and an excellent exposition of local stability and phase diagram analysis.
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 that informal local stability test for the CP model with static expectations – viz. (2-18) – 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.
Figure 2.7: Global stability with forward-looking expectations and high migration costs
Global Stability
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, the transversality condition). 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: (1) it is much easier to find the unstable saddle path than the stable saddle path, and (2) the stable
sn W
CPN
CPS
sym
path becomes the unstable path in reverse time.30 Numerical techniques are also used to solve the second source of intractability, namely the fact that ω-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.31
Figure 2.8: Global stability with intermediate migration costs
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 considered 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 2.7. 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 2.8, is for an intermediate value of
migration costs. Here the saddle paths overlap somewhat since the Jacobian evaluated
30 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.
31 Algorithms showing how to find saddles paths and approximate Ω[sn] are available from the web site http://heiwww.unige.ch/~baldwin/.
sn W
CPN
CPS
sym A B
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 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 equa lly rational for them to jump on the track that will take them to the CPN outcome.
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 choose 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 that 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
(
ρ± ρ2 −4(dΩ/dsL)sL(1−sL)/γ)
/2, so we get complex roots when migration costs are sufficiently low, namely when:(2-24) 4( / )2 (1 )
γ < dΩ dsρL sL −sL
To summarise, the possibility of history- versus-expectations dynamics, i.e.
that ‘self- fulfilling prophecies’, or self- fulfilling changes in expectations arises when the costs of migration (i.e. γ) 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 2.9, 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.32
32 Karp (2000) qualifies this insight by assuming that agents have ‘almost common knowledge’ in the sense of Rubinstein (1989) rather than common knowledge about history (economic fundamentals). In
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.
To sum up, when migration costs are sufficiently low, and φB<φ<φS, the symmetric equilibrium can be globally unstable while being locally stable and a coordinated change in expectations could produce migration that would shift the economy from the symmetric equilibrium to a CP outcome even though there were no change in parameters or trade costs. In other words, a self- fulfilling prophecy could break the symmetric outcome even though it is locally stable.
Figure 2.9: Global stability with low migration costs