This Appendix proves the following:
Proposition: For any initial values of q and b that satisfy (25b) and (25d),
Proof: (a) Suppose that
which implies, given that μ1 < μ2, that
The Case (b) can be dismissed for the three dynamic exercises conducted in the text because of the following:
(1) In case of devaluation or a decrease in government spending, case (b) is inconsistent with b being constant across steady states.
(2) In the case of a change in β, given that
, M and b would move in opposite directions which is inconsistent with their new steady-state values.
We now show how the initial jump in ρ can be established. Recall (21) given by:
Consider the case of a devaluation. We have already established that, q falls on impact which means that the last term on the right-hand-side of (21) falls on impact. On the other hand,
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This paper was initiated while Carlos Végh was in the Research Department. The authors acknowledge helpful comments by Guillermo Calvo, Pablo Guidotti, Mohsin Khan, and Jonathan Ostry on an earlier draft of this paper. The usual disclaimer applies with respect to any remaining errors.
The possibility of inter-market transactions is briefly noted in Flood and Marion (1988). Tornell (1988) also recognizes the possibility of leakage; however, the model he uses abstracts from consumption-savings decisions and illegal transactions are carried out by risk-neutral investors.
In non-optimizing models such as Lizondo (1987), for example, a steady-state spread may be indefinitely maintained.
The foreign price level is assumed to be constant so that both the nominal and real quantities of foreign bonds in the hands of domestic residents is fixed.
As pointed out by Obstfeld (1986), the fact that reserves earn interest implies that, under fixed exchange rates, devaluation is neutral.
It is assumed, as it is usually the case, that the Central Bank does not monetize the nominal capital gains on reserves but instead creates a fictitious non-monetary liability.
Note that, even if consumption rises over time, it is initially at a lower level than before the disturbance.
The behavior of q will be dramatically altered when leakages between the commercial and financial markets are incorporated, as will be shown in the next Section.
As noted by Obstfeld (1986), the fact that steady-state consumption is higher does not mean that a devaluation is welfare improving. To the contrary, the losses in utility attributable to diminished consumption in the short run outweigh the gains from a higher steady-state consumption. This follows from the fact that, for given resources, a constant path of consumption is preferable to a non-constant path.
This functional form permits considerable simplicity in that the resulting expressions are linear.
Note that the present model is not stochastic and as such, there is no uncertainty regarding the eventual detection by the authorities of the fraudulent activities. It will also be assumed that γ, the fraction of bonds confiscated takes the linear form γ(qb) = βqb, where β is a constant which is small enough to ensure that γ remains below unity. Thus, the larger the amount of bonds in the possession of the trader, the higher the portion of bonds that is forfeited.
A more natural assumption would be that the confiscation carried out by the authorities applies only to those bonds illegally acquired. This assumption, however, would render the model analytically intractable. While the assumptions just spelled out are undoubtedly special ones, they are needed for analytical tractability. If these assumptions were not made, it would be necessary to resort to numerical methods which are necessarily limited in generality.
Recall that this condition is necessary in the no-leakage model, which is a particular case of the present one, to ensure existence of the only negative characteristic root.
For instance, if hil = 0 then (28a) implies that (1-φ) hi3 = rhi4. Substituting the latter into (28c) yields (r-μi)hi4 = 0 whence it follows that hi4 = 0. This in turn implies, from (28a) that hi3 = 0 which (from (28b)) implies that hi2 = 0. The other cases are straightforward.
In principle, there are many possible paths from A to B in the (b, M) plane but, based on Figure III, we can infer that the path depicted in Figure IV is in fact the actual adjustment path. Figure III shows that b can follow only two possible paths. Since, when φ = 1, this model degenerates into the complete separation model studied in the previous section, in which case a devaluation causes a fall in q, it should be clear that, at least for high values of φ, the same happens under incomplete market separation. Therefore, in the case of a devaluation, the path in Figure III, going from A to 0, is the relevant one.
Naturally, in the particular case where r=δ, a change in β does not affect the steady state because b=0 is the only level of bond holdings that is consistent with that parameter configuration.