1. The Devaluation rule

The central bank, having fixed the exchange rate at ē at time t, must decide at the end of period t+1, on the basis of some viability criterion, whether or not to adhere to the official parity. We assume in what follows that the authorities will maintain the fixed exchange rate prevailing at t as long as the level of the parallel market premium is not “too high”. If, however, the premium rises above a critical upper bound during t+1, $\overline{\rho }$, the official exchange rate will be devalued at the end of that period, and the central bank will establish a new fixed exchange rate, $\tilde{e}$_t+1 using a time-invariant policy rule.

The rule for setting the post-devaluation exchange rate will in general be a function of the structure of the economy. ^6/ As the new fixed exchange rate that would be set after a devaluation, $\tilde{e}$_t+1 must be “viable” in the following sense: it must be such that the condition ${\tilde{\rho }}_{t+1}\le \overline{\rho }$ is satisfied (${\tilde{\rho }}_{t+1}$ denoting the post-collapse premium), since otherwise the post-devaluation official rate is not sustainable. A particular rule that fulfills this condition is

Since the coefficient δ is positive, the devaluation rule described by equation (1) produces a post-devaluation exchange rate greater than the minimum viable rate, that is, the rate required to set ${\tilde{\rho }}_{t+1}$ = $\overline{\rho }$. This implies that the post-devaluation premium falls below the critical upper bound. ^7/

Several rationales can be provided for a devaluation rule of the type described by equation (1): concerns over the degree of under-invoicing of exports/over-invoicing of imports, the level of the real exchange rate, or the distorsions created by a widening differential between the official and parallel exchange rates. An alternative mechanism that may trigger the rule is the inflation tax, as discussed below.

2. Prices and the Money Market

The level of the premium given in the rule (1), is determined by the structure of the economy. Formally, the equations describing the equilibrium of the money market and domestic prices are given by the following log-linear system:

where m_t denotes the nominal money stock, d_t domestic credit, the p_t domestic price level, $\overline{R}$ the stock of net foreign assets held by the central bank, e_t the officially fixed exchange rate, and b_t the parallel market rate. All variables are measured in logarithms, except ε_t which denotes a random variable with zero mean, whose distribution is specified below. E_tZ_t+1 denotes the value of z agents expect to prevail at the end of period t+1, given information available at the end of period t.

Equation (2) describes the money market equilibrium relationship in the presence of currency substitution and foreign exchange rationing. The demand for domestic currency is inversely related to the (one-period ahead) expected rate of depreciation of the parallel market exchange rate. ^8/ Equation (3) is based on a log-linear approximation and defines nominal money supply as the sum of net foreign assets (which are constant in foreign currency terms since the authorities allow no changes in reserves) ^9/ and domestic credit. Equation (4) shows that the rate of growth of domestic credit is explained by deterministic factors (μ, which reflects the need to finance the fiscal deficit), and by unpredictable elements, captured by the random variable ε_t. Although unknown to agents at the beginning of the period, the credit shock takes a realized value during period t.

Equation (5) indicates that the price level depends on the official exchange rate and the level of the premium, defined as the difference between the parallel rate and the official exchange rate. This results from the assumption that trade takes place through both official and illegal channels, with the price of smuggled imports reflecting the marginal cost of foreign exchange—that is, the parallel market exchange rate. Equation (5) derives from the assumption of purchasing power parity at a composite exchange rate, with the foreign price level set to unity for simplicity. ^10/ Since the domestic and foreign good are perfectly substitutable, marginal cost pricing implies that σ should be close to 1. ^11/

3. The Premium and the Inflation Tax

Before deriving a behavioral equation for the parallel market exchange rate from the above system, the process by which agents assess the viability of the official exchange rate must first be described. If, in agents’ minds, the official exchange rate is pegged at a “credible level”, its expected rate of depreciation will always be zero. However, agents do not necessarily believe that the official parity will be adhered to by the authorities in all circumstances. Instead, when forming expectations about the next period’s official exchange rate, they will usually assess the credibility of the current parity, based on the information available. A convenient and fairly general formulation is to postulate that the degree of credibility of the official exchange rate is inversely related to the current level of the premium. When the premium is perceived as being “too high”, the fixed exchange rate will appear impossible to defend, and the credibility of the official parity will be undermined. Formally, it may be assumed that agents attach the probability _tq to the eventuality of a devaluation occuring at the end of the next period, and the probability 1 - _tq to adherence to the current parity. If a devaluation occurs, agents expect a depreciation of the official exchange rate by a proportion η, which is taken as constant for the monent. The rate of depreciation of the official exchange rate expected by the public between the end of period t and t+1 is therefore given by

The devaluation probability (or, equivalently, the degree of credibility of the official parity) _tq formed by portfolio holders may depend, in general, on a number of factors. The crucial feature of the model is that _tq is taken to be an increasing function of the premium, that is, ∂_tq/∂ρ_t > 0. If agents are fully confident that the current official parity will be adhered to, then _tq = 0, which occurs only for ρ_t = 0. In addition, it is also assumed that _tq_t+k = _tq ∀k = 1, 2,… ∞ (the devaluation probability is independent of the expectational horizon), and that lim_tq = 1 for ρ_t → ∞. ^12/

From equations (2), (3) and (5),

where c₀ = (a₀ - υ$\overline{R}$)/(1 - υ), and c₁ = a₁/(1 - υ).

A convenient procedure to find a solution to this equation is the method of undetermined coefficients. Conjecturing the solution form b_t = λ₀ + λ₁d_t + λ₂e_t, and substituting this trial solution in (7) yields, using (3) and (6),

where c = c₁ [μ] + _tqη(s - 1)/s - c₀ and s = σ/(1-υ) > 1, since by assumption σ is close to 1 and υ < 1. Defining the parallel market premium ρ_t as the difference between the parallel and official exchange rate yields

Equations (8a) and (8b) describe the evolution of the parallel exchange rate and the premium when agents attach a probability _tq to a devaluation of the official exchange rate occuring next period. The higher this probability is, the higher will be b_t and ρ_t. For given expectations, a rise in the stock of domestic currency (through credit expansion) is—for a given official exchange rate—associated with a rise in the free exchange rate which reduces the expected rate of depreciation of the unofficial price of foreign exchange, and helps restore portfolio equilibrium. From equation (8a),

which implies that an increase in the rate of expansion of domestic credit raises, by the proportion Γ/s = [1 + η(s - 1) (∂_t q/∂μ)]/s, ^13/ the expected rate of depreciation of the parallel exchange rate. As a consequence, the demand for domestic currency falls. Credit growth and the rise in the parallel rate raise anticipations of a future increase in the free exchange rate, reduce the credibility of the current official exchange rate, and thereby cause a reduction in domestic currency holdings.

The above description of the relationship between an expansion in domestic credit and the parallel market premium has an important bearing on what happens to government revenues from the issuance of money, namely the inflation tax. Assume for the moment that ε_t = 0 ∀ t. The yield of the inflation tax—assuming no devaluation takes place—is given by the product of the rate of domestic money growth, (1 - υ)μ, and the real money stock. Since, as shown above, an increase in the rate of expansion of domestic credit is associated with a fall in the real stock of domestic currency held in private agents’ portfolios, (the logarithm of) the proceeds from the inflation tax will fall or rise according to whether the derivative of log[(1 - υ)μ] + m_t - p_t = log[(1 - υ)μ] + a₀ - a₁ [μ + _tqη(s - 1)]/s, with respect to μ is positive or negative. For a₁ > s/μΓ, that is, for a high enough degree of currency substitution, inflation tax revenue may actually fall as a result of an expansionary credit policy. The higher the value of μ is, the more likely it is that the above condition will be satisfied.

The general implication of this result is that the higher the rate of expansion of the nominal stock of domestic credit, the higher will be the expected rate of depreciation of the parallel exchange rate, and, consequently, the lower the proportion of domestic currency held in agents’ portfolios. As the public shifts away from domestic to foreign currency, the relative importance of the monetary base, and hence the financing that can be achieved through money creation, is reduced for any given rate of growth of domestic credit. If the degree of currency substitution is “high” enough, the gain in revenue from accelerating credit growth is more than offset by the erosion of the tax base that results from the decline in holdings of real balances, due to the now higher expected rate of depreciation of the parallel market exchange rate. As a consequence, the flow of seignorage from the inflation tax falls. ^14/

What can the government do to prevent the erosion of its inflation tax base? Obviously, the “first best” strategy is to reduce the rate of growth of domestic credit by a sufficient amount to ensure that the condition μ < s/a₁Γ is satisfied. This may not be, however, a feasible policy option in the presence of fiscal rigidities. The key assumption made in this paper is that when, as a result of a sustained expansionary credit policy, revenues from the inflation tax have fallen to a critical level, the authorities may devalue the official exchange rate—using the rule given by equation (1)—in an attempt to reverse the process. From equations (8a), (8b) and (9), a once-and-for - all devaluation implemented at the end of period t implies a depreciation of the parallel exchange rate (in the proportion [s - 1]/s), a fall in the level of the premium (in the proportion 1/s) and in the probability that a devaluation will occur in the following period, and therefore a fall in the anticipated rate of return on foreign currency holdings. The devaluation “works” by (partly) restoring the credibility of the fixed exchange rate regime, and this reduces the expected rate of return on foreign money for the next period, since expectations are formed at the end of period t. This provides an incentive to switch back into domestic currency assets. Domestic money balances will increase (by the proportion a₁η(s - 1)[∂_tq/∂ρ_t]), and the yield of the inflation tax will rise. However, the fall in the expected rate of depreciation of the parallel exchange rate will only be temporary since, in the next period, the level of the premium will resume its upward course, if monetary and fiscal policies are unaltered, implying that the devaluation probability formed by asset holders will rise again. A once-and-for-all devaluation, therefore, can only provide a temporary increase in the tax base, a proposition whose implications are further discussed below.

In a country where domestic residents can hold foreign money balances, currency substitution can become an effective way of avoiding the inflation tax on the holdings of domestic cash balances. If, at the same time, government deficits are high, the notion that the authorities may have to protect—or recapture, through a devaluation—its inflation tax base has some credence. ^15/ An alternative rationale for government intervention, which can be referred to as a “sovereignty” argument, is to assume that the government will not permit the complete collapse of its currency. This implies that although there exists a rate of domestic credit expansion—and, therefore, an expected rate of depreciation—that is high enough to drive agents to liquidate the whole stock of domestic money in their portfolios, in practice agents will not be able to substitute away completely from the domestic currency. Through institutional constraints (legal requirements that domestic money be used for specific transactions, for example), the authorities can prevent the national currency to be driven out of existence. In this context, a devaluation may be viewed as an alternative means to prevent a complete “dollarization” of the economy. Although this argument is also quite appealing, in what follows we will stress the inflation tax mechanism as the major factor underlying the decision to devalue.

In light of the foregoing discussion, it will be assumed that the official exchange rate is maintained only if the proceeds from the inflation tax do not fall as a result of an increase in the rate of growth of domestic credit. ^16/ Generalizing the condition given above, the actual yield of the inflation tax (which is known only after the realization of the domestic credit shock) rises following a credit expansion only if μ + ε_t+1 < s/a₁Γ. Since, from equation (8b), the actual rate of change of the premium between t and t+1—assuming no devaluation takes place in the interval, so that e_t+1 = e_t = ē—is equal to (μ + ε_t+1)/^s> the above condition is equivalent to ρ_t+1 - ρ_t < 1/a₁Γ.

Concerns over the yield of the inflation tax imply, therefore, that the authorities will maintain the fixed exchange rate prevailing at t as long as the actual rate of increase of the parallel market premium is not “too large”. If, however, the rate of change of the premium rises above a critical upper bound during t+1 (implying an unacceptable fall in the yield of the inflation tax), the official exchange rate will be devalued at the end of that period, and the central bank will establish a new fixed exchange rate. A devaluation implemented at the end of t+1 will lower the level of the premium, and (since expectations are formed at the end of every period) will reduce the probability of a futher devaluation occuring next period as well as the expected rate of depreciation of the parallel exchange rate. This will imply an instantaneous shift away from foreign currency holdings at the end of t+1, as discussed previously.

Since the value of the premium at t is common knowledge at the end of period t, a viability criterion which states that the rate of increase of ρ between t and t+1 should be lower than a given value is equivalent to the requirement that the level of the premium during t+1 remains below a given upper bound, $\overline{\rho }$. ^17/ However, although agents may know $\overline{\rho }$ with certainty (or, more generally, they may form a prior probability distribution over the possible values that $\overline{\rho }$ may take on), the exact timing of the collapse of the official parity cannot be predicted at t, because of the stochastic nature of the credit shock.

The devaluation rule (1) ensures that the post-collapse premium falls below the critical upper bound, implying a fall in the devaluation probability formed by asset holders and in the expected rate of depreciation of the parallel rate for t+2, a rise in domestic currency holdings, and an increase in revenue from the inflation tax.

1. The Devaluation Probability

As argued in the previous section, if the officially fixed exchange rate proves to be unsustainable due to the rise in the premium, the authorities will devalue the domestic currency in an attempt to maintain the yield of the inflation tax. Let _tπ_t+1 denote the probability, as of the end of period t, that the official exchange rate will be devalued at the end of t+1. ^18/ The probability that the current exchange rate will be maintained is therefore given by (1 - _tπ_t+1). The unconditional expected official exchange rate at t+1 can therefore be expressed as

where E_t(ē_t+1 |ρ_t+1 > $\overline{\rho }$) denotes the exchange rate that would prevail if the authorities devalue at the end of period t+1.

The probability as of the end of period t that a devaluation will occur at the end of period t+1 is simply the probability that, under the current parity, the parallel market premium rises above $\overline{\rho }$ during t+1. From equation (8b), this probability depends in turn on the probability that a sufficiently large realization for domestic credit occurs. Formally,

or, using (4),

where now c = c₁[μ + _tqη(s - 1)]/s - c₀ - ē. The credit shock ε_t is assumed to obey

where 1/λ² denotes the variance of ε_t, and ν_t a random variable distributed exponentially with an unconditional probability density function: ^19/

To ensure that the rate of growth of domestic credit remains positive, we will assume that μ > 1/λ. Using (12) and (13), we can write (11’), using the variable k_t for notational ease, as

where k_t = s$\overline{\rho }$ - (c + μ) + 1/λ - d_t, and F(.) is the cumulative distribution function associated with f_ν (ν_t). Using the density function (13), the cumulative probability of a devaluation is therefore

Integrating (15) yields:

Equation (16) shows that the higher the upper limit on the premium $\overline{\rho }$ is, or the higher σ (since ds/dσ > 0), the lower the probability of a devaluation. A rise in σ, which measures the sensitivity of the price level to changes in the premium, reduces the impact of the expected rate of depreciation of the parallel market rate, dampens the shift away from domestic currency, and reduces therefore the effect of a random increase in domestic credit growth on the premium. By contrast, the higher the rate of expansion of domestic credit, the higher the probability of a collapse of the fixed rate. The collapse probability depends also on the degree of credibility asset holders attach to the official exchange rate. Specifically, it is easy to shov. from (16) that ∂_tπ_t+1/∂_tq > 0: the lower the degree of credibility of the official parity (the higher _tq), the higher will be the economy-wide probability of a devaluation occuring at the end of period t+1. ^20/ A devaluation occurs with certainty if k_t < 0. ^21/ For k_t ≥ 0, the probability of a collapse can be completely eliminated only when λ → +∞. Since 1/λ² is the variance of the exponentially distributed random domestic growth term, this condition implies that agents must know with certainty the credit rule.

The solution procedure leading to equation (16) can be used to calculate not only the probability of a parity change occuring at the end of period t+1 but also, more generally, the probability at t that the fixed exchange rate regime collapses in period t+n (neither earlier nor later) and the probability, at t, that it collapses no later than period t+n. It can be shown that the devaluation probability increases with the length of the interval (t,t+n), and that whatever the length of this interval, a rise in the deterministic component of the rate of growth of domestic credit always increases the probability of an exchange rate crisis. The proofs of these propositions are contained in Appendix I.

Finally, an important aspect of the solution (16) is that the devaluation probability is independent of the devaluation rule. As shown below, this depends critically on the assumption that asset holders do not know the policy reaction function of the authorities and/or do not use that information in forming their expectations about the future official exchange rate.

2. The Conditional Exchange Rate

Consider now the determination of the conditional exchange rate expected to prevail at the end of period t+1. From the above discussion, E_t($\tilde{e}$_t+1|ρ_t+1 > $\overline{\rho }$) = E_t($\tilde{e}$_t+1|υ_t+1 > k_t). Using the devaluation rule (1) and equations (4) and (8b), the conditional expectation can be expressed as

To find E_t(υ_t+1|υ_t+1 > k_t, we f°rm tne conditional probability density function over υ_t+, where the conditioning information is υ_t+1 > k_t. As shown by Flood and Garber (1984, pp. 12-13), this conditional density function is:

Hence, for k_t ≥ 0,

Substituting this result in (17) yields

or, using the definition of k_t,

Equation (21) shows that the higher δ, the higher will be the post-devaluation exchange rate, and the lower the post-devaluation premium. Also, the higher the elasticity of domestic prices with respect to the premium, the lower will be the post-collapse exchange rate. However, the post-devaluation exchange rate does not depend on the degree of credibility attached by asset holders to the existing official parity.

Finally, substituting (21) in (10) and using (16) yields the unconditional forecast of the exchange rate for t+1:

The unconditional expected exchange rate will depend on all factors affecting the devaluation probability and the post-collapse, conditional exchange rate.

If a collapse indeed occurs, the premium will fall, ^22/ generating a temporary increase in domestic currency holdings. If credit policy remains expansionary, however, the premium will resume its upward movement, eventually leading to another “forced” devaluation. The initial drop and subsequent rise in the premium following a devaluation is consistent with the empirical evidence provided by Edwards (1989) and Kamin (1988) on the behavior of parallel markets for foreign exchange in developing countries.

1. Consistent Devaluation Expectations

It has been assumed in the previous discussion that agents, uncertain about whether the fixed exchange rate will be adhered to, learn over time about the viability of the official parity by observing the behavior of the authorities and the parallel market premium, and infer whether domestic credit and fiscal policies are consistent with the current exchange rate. At the end of each period, agents make their portfolio decisions based on their assessment of the probability that the official exchange rate will be devalued at the end of the following period. Their actions, in turn, affect the economy-wide probabilty of a devaluation.

Consider now the case where expectations about the future rate of devaluation of the official exchange rate are not exogenously given, but depend on agents’ expectations about the monetary authorities’ policy reaction function. That is, agents know the exchange rate rule (9) and use this information to form consistent predictions. The expected rate of depreciation of the official exchange rate is now given by

where 0 < _tq < 1, measures the degree of credibility agents attach to the official parity. ^23/ By the definition of ρ_t,

E_tb_t+1 - b_t = E_tρ_t+1 - ρ_t + _tη_t+1.

Combining this expression with (23) yields

Equation (23’) shows that, with endogenous expectations, a rise in the level of the premium affects the rate of devaluation expected by asset holders through two channels: the effect of ρ_t on _tq (and therefore $\overline{\tau }$) as before, and the effect of ρ_t on the rate of devaluation agents expect the authorities to implement, should they decide to devalue. A rise in the expected rate of depreciation of the parallel exchange rate raises also the expected rate of depreciation of the official rate, but less than proportionally ($\overline{\tau }$ < 1).

Using (2)-(5) and (23’), and positing as before the solution form b_t = λ₀ + λ₁d_t + λ₂e_t, it can be shown that the coefficients λ_k are solution of the following system:

In the particular case of “full” credibility by asset holders (_tq = 0) > equations (8a) and (24a)–(24c) yield the same solution for the parallel market exchange rate. In the general case, however, (24a)-(24c) form a non-linear system in the coefficicients λ_k, with, λ₀ and λ₁ solving residually once λ₂ is determined from equation (24c). The quadratic equation (24c), given that its discriminant Δ is positive, has two real and positive roots (by Descartes’ rule of signs), given by

λ₂ - [($\overline{\tau }$(s + C₁ - 1) + s) ± Δ^1/2]/2$\overline{\tau }$(s + c₁).

However, it can easily be shown that both roots cannot be lower than unity. Consistency between the properties of the model and “stylized facts” requires selection of the only root that is lower than one, denoted λ₂. ^24/ Simple manipulations yield now

implying that as before, a devaluation, by reducing the level of the premium and lowering expectations of a future devaluation, reduces the expected rate of return on foreign currency holdings.

Using equations (24a)–(24c), the devaluation probability is now given by

where ($k_t=\left(\overline{\rho }-{\tilde{\lambda }}_0-{\tilde{\lambda }}_2\overline{e}\right)/{\tilde{\lambda }}_1-\left(\mu +d_t\right)$, and $\tilde{\lambda }$₀ and ${\tilde{\lambda }}_0\mathrm{and}{\tilde{\lambda }}_1$ denote the values of λ₀ and λ₁ associated with $\tilde{\lambda }$₂. As before, the collapse probability depends on the degree of confidence agents attach to the official exchange rate. It is tedious but straightforward to show from (25) that ∂_tπ_t+1/∂_tq > 0 : the lower the degree of credibility of the official parity, the higher will be the probability of a devaluation at the end of period t+1. In addition, the devaluation probability depends also on the parameter characterizing the devaluation rule, the coefficient δ. The higher the rate of devaluation in the policy rule, (the higher is δ), the higher the likelihood that a parity change will occur at the end of t+1, because of its induced effect on expectations formed by asset holders.

2. Uncertain Central Bank Policy Response

Up to this point it has been assumed that a devaluation is known to be imposed with certainty when the premium becomes greater than $\overline{\rho }$. The analysis, however, can be extended to deal with situations in which the policymaker’s reponse is uncertain. ^25/ Two cases are considered here. First, uncertainty relative to the value of the premium that triggers the devaluation; and second, uncertainty regarding the parity change itself, as well as its size.

A fairly general way to introduce uncertainty regarding the critical upper bound is to consider the case where the central bank randomly changes the value of the premium that will trigger the devaluation. This has important implications concerning the timing of the parity change. For a given growth rate of domestic credit, even though agents may know the general form of the authorities’ decision rule, they cannot calculate with certainty the critical level of the premium that will result in a discrete parity change. Formally, the probability of a devaluation becomes

where ζ_t+1 = 1/ϕ + ω_t+1 denotes a random variable distributed with zero mean and variance 1/ϕ², and ωt is distributed exponentially with mean 1/ϕ and density ^26/

Using (4), (8b) and (27), equation (26) can be written as

where now k_t = s$\overline{\rho }$ - (c + μ) + 1/σ - d_t - s/ϕ. To evaluate the cumulative probability of a collapse requires now to set up a convolution integral. Let χ_t+1 = sω_t+1 = (1/s)f_ω (X_t+/s), so that f_χ(χ_t+1) = (1/s)f_ω(χ_t+1/s), and define z_t+1 = υ_t+1 - χ_t+1. If υ_t+1 and ω_t+1 are independent random processes, the probability density function of z_t+1 is (see for example Kendall and Stuart, 1977, p, 280):

which becomes, after integrating,

The collapse probability (28) can therefore be written as

Integration yields

Equation (31) shows that the higher the degree of uncertainty on the level of the premium that triggers the devaluation rule, that is, the lower ϕ is, the higher the probability of the devaluation (∂_tπ_t+1/∂ϕ < 0).

As an alternative way to introduce uncertainty over policy response in the model, consider now the case where the upper bound $\overline{\rho }$ is perfectly known to the public. Suppose further that the public is uncertain about whether, if the viability condition is violated (that is, ρ_t+ ≥ $\overline{\rho }$), the central bank will effectively devalue or not at the end of period t+1. Formally, agents consider that a devaluation will be imposed with probability α. ^27/ Then the exchange rate the public expects to prevail at the end of t+1 is

or, using (21)

The unconditional exchange rate will now depend not only on the distribution of the domestic credit shock, the level of domestic credit and the official exchange rate, but also on the probability that the public attaches to the authorities’ intentions to devalue—if they have to. The higher α is—that is, the higher the probability that a devaluation will effectively take place if the premium overshoots its viable level—the higher will be the expected post-collapse exchange rate.

Finally, one can consider uncertainty about the size of the devaluation—that is, uncertainty about δ. Without recourse to formal derivations, it is intuitively clear that under the assumptions of the basic model, stochastic movements in δ have no effect on the devaluation probability if the rate of depreciation of the official parity asset holders expect to prevail in the event of a devaluation is exogenous—although this does not hold if devaluation expectations are consistent with the devaluation rule—and will also have no effect on the conditional exchange rate if they are independent of credit shocks.

3. Controls and the Postponment of the Crisis

Consider now the case where the authorities try to postpone the parity change and intensify foreign exchange controls when the premium reaches the level ρ_min. ^28/ For convenience, let us assume that the central bank will maintain the fixed exchange rate as long as the premium remains in the interval (ρ_min, ρ_max), where ρ_max > ρ_min, and will devalue (with probability one) only when ρ_t+1 > ρ_max, using rule (1).

Although exchange restrictions have not been explictly modeled here, a heuristic discussion of the issue may be sufficient to describe the process at work. An intensification of controls—measures, for instance, that tend to make it more costly to substitute foreign for domestic money in the official market—is likely to increase excess demand for foreign currency at the official exchange rate, with a consequent spill-over effect on the parallel market. The result will be a depreciation of the free exchange rate, an increase in the premium, a rise in the devaluation probability formed by asset holders, and a rise in the expected rate of return on foreign currency holdings which will hasten—assuming credit policy remains unchanged—the attainment of the upper limit ρ_max. ^29/ Therefore, in this model, a reinforcement of foreign exchange controls in an effort to postpone exchange rate or fiscal adjustment only makes matters worse, by increasing the likelihood that a “forced” devaluation will eventually take place. The “credibility effect” created by the rise in the level of the premium defeats the very purpose of the controls.

4. Endogenous Credit Policy Switches

As is typical in models of balance of payments crises where credit policy is exogenous, the basic framework of this paper shows that the devaluation crisis is the outcome of excessive monetary expansion. ^30/ The analysis implies that a fixed exchange rate regime cannot be viable indefinitely—although the transition to a new official exchange rate is diffuse, due to the stochastic nature of the credit growth process. A further implication of the model is that as long as credit policy follows an expansionary path, and as long as the authorities maintain their objective of not allowing the premium to rise above a critical bound (that is, as long as the expected yield of the inflation tax is not allowed to fall below a critical level), the economy will undergo a succession of devaluation crises. ^31/ Typically, the premium will fall following the devaluation—implying a temporary switch away from foreign currency holdings—and then will resume its upward movement, until a new devaluation takes place.

The apparently ineluctable character of a devaluation crisis, as well as its recurrent nature, runs into the same kind of conceptual difficulties encountered in models of collapsing fixed exchange rate regimes in which the “excessive” rate of domestic credit creation is the result of an exogenously determined fiscal deficit; namely, why is it that the authorities do not attempt to prevent the crisis by adjusting their fiscal and credit policies? ^32/ Indeed, the probability of collapse derived earlier is non-zero simply because it has been assumed that the authorities maintain some flexibility with respect to domestic credit creation. Fiscal discipline, that is, a reduction in unpredictable changes in credit behavior, can significantly reduce the-devaluation probability.

Moreover, there is nothing in the model “forcing” the central bank to devalue at the moment the premium reaches its critical upper bound; instead, the authorities could as well change their monetary policy rule to ensure viability of the official parity. Some recent models of balance of payments crises have considered this type of endogenous changes in monetary policy. ^33/ For Drazen and Helpman (1988) for instance, the assumption that the authorities initially choose the exchange rate as a stabilization instrument, instead of altering underlying macroeconomic policy that is inconsistent with the existing exchange parity, may serve as a first step in a two-stage process. But ultimately, if the new exchange rate regime is inconsistent with the underlying policy process, there will be a need for a new policy regime. A similar argument can be made in the case considered here.

One simple way to introduce endogenous credit policy adjustment is to consider, instead of (4), a credit rule with feedback: ^34/

according to which the authorities vary the average rate of growth of domestic credit in t+1 inversely with the level of the premium at t. The solution procedure is now slightly more involved. Positing as before a solution of the form b_t - λ₀ + λ₁d_t + λ₂e_t, and using (2)-(5) and (33) to solve with respect to the premium, it is easily shown that the coefficients are solutions of the following system:

Equations (34b) and (34c) solve for λ₂ and λ₀ residually once λ₁ is determined from (34a). Because equation (34a) is quadratic, and since its discriminant is positive, there are two real and distinct possible values for λ₁, given by

By Descartes’ rule of signs, one root is positive and the other negative. A simple way to choose among the two roots is to consider the case where γ → 0—the “no-feedback” rule. In this case, 2c₁γλ → 0, and (σ₂ + 4c₁γ)^1/2 → σ. Therefore, for the above equality to be satisfied when γ → 0, the only meaningful solution is the positive root, denoted below $\tilde{\lambda }$₁. It is easy to see from (35) that d$\tilde{\lambda }$₁/dγ < 0, so that a higher feedback coefficient reduces the impact of the rate of growth of domestic credit on the premium.

Solving the basic model as before yields

where now $k_t=\left(\overline{\rho }-{\tilde{\lambda }}_0-{\tilde{\lambda }}_2\overline{e}\right)/{\tilde{\lambda }}_1-\left(d_t+\mu +{\mathit{\gamma \rho }}_t\right)$, and ${\tilde{\lambda }}_0\mathrm{and}{\tilde{\lambda }}_1$ denote the values of λ₀ and λ₂ associated with the positive root of (34a). Again, tedious calculations indicate that ∂_tπ_t+1/∂γ < 0; the higher the feedback coefficient is, the lower the probability of a devaluation. In the “perfect feedback” case, that is, for γ → ∞, the devaluation probability falls to zero. But now, for a given $\overline{\rho }$, the impact of the premium on the collapse probability is ambiguous. A rise in ρ_t has a negative impact on d_t1, which reduces the possibility that the next realization of credit growth forces the devaluation. But a rise in ρ_t also lowers the credibility of the current fixed exchange rate now and raises the expected rate of return on foreign currency holdings, which gives agents an incentive to shift away from domestic currency, and increases the likelihood that the authorities will implement a devaluation at the end of t+1.