A. General Assumptions

The domestic country is assumed to be small in world markets, allowing it to take foreign variables as constant. To keep the model simple, both purchasing power parity and interest rate parity are assumed, yielding, respectively:

where P is the domestic price level, S is the spot exchange rate, defined as the domestic price of foreign currency, i is the nominal interest rate, and an asterisk (*) denotes a foreign variable. For convenience, the foreign rate of inflation and money growth are assumed to be zero, and the foreign price level is normalized at unity. Additionally, the foreign interest rate is assumed to equal the rate of time preference of the representative domestic agent (θ), assuring the existence of a steady state equilibrium.

B. Monetary and Fiscal Policy

With the simplification of no banking system, the money supply (M) is determined by the sum of domestic credit (D) and interest-bearing foreign exchange reserves (SF). Therefore, money and monetary growth are given, respectively, by:

where a dot (·) denotes a time derivative. The overbar on S denotes the fixed exchange rate. The authorities also intervene in the foreign exchange market in the event of a discrete speculative attack on reserves, allowing money to change discretely according to:

Fiscal policy is defined very simply. The government (consolidated monetary and fiscal authority) obtains revenue from domestic credit creation$\left(\dot{D}\right)$ and from interest earnings on assets (θSF) ⁴ It transfers these revenues to the public in a lump-sum manner according to:

where τ is the real value of transfers.

Combining equations (2) and (3) yields an expression for the government’s budget constraint:

where $\mu =\frac{\dot{M}}{M}$, the nominal rate of money growth. The government accumulates foreign exchange reserves as transfers fall short of interest on existing government assets and seigniorage revenues. Under the assumption that monetary growth in the rest of the world is zero (μ^* = 0), fixed exchange rates imply that domestic monetary growth is zero (μ = 0). Equation (4), therefore, determines the change in foreign exchange reserves. Under flexible rates, μ is given by the value which sets $\dot{F}=0$ in equation (4).⁵

C. Money Market Equilibrium with Forward-looking Agents

Consider, now, private sector behavior. Assume that the small open economy is populated by representative agents with identical logarithmic utility functions containing consumption and real money balances. A representative agent faces a budget constraint whereby the real value of the agent’s wealth (a), which consists of real bonds (b) and the real value of nominal money balances $\left(\frac{M}{S}\right)$, accumulates over time as the agent receives interest on wealth, transfers from the government (τ), and endowments (y), and falls with expenditures on consumption (c) and real money balances $\left(i\frac{M}{S}\right)$ according to:

where i is the nominal interest rate.

Assuming that the future stream of endowments is constant and that the domestic rate of time preference equals the world real interest rate, maximization of utility subject to the budget constraint and to the “no Ponzi game” constraint, given by $\underset{t\to \infty }{{lim}}{a}_t{e}^{-\theta t}=0$ yields consumption demand and real money demand equations according to:

Consumption’s share in total expenditures is given by 1−γ, and ψ is the present discounted value of government transfer payments. Total expenditure on consumption and real money balances $\left(c+i\frac{M}{S}\right)$ is proportional to total wealth, defined as financial wealth $\left(\frac{M}{S}+b\right)$ plus the present discounted value of disposable income $\left(\frac{y}{\theta }+\psi \right)$. Additionally, expenditures on money $\left(i\frac{M}{S}\right)$ are proportional to expenditures on consumption. Since interest equals time preference, consumption is constant over time.

Using (5) and (1) in (6) yields the money market equilibrium equation as a function of the exchange rate and its rate of change.

This equation and equation (4) are the fundamental equations of the model. Monetary equilibrium differs from that in the standard model in two ways. First, the scale variable, determining money demand, contains forward looking variables. The expression $\frac{y}{\theta }+\psi$ denotes the present value of future income and transfers. Second, the government’s budget constraint, equation (4), implies that τ, ψ, and μ cannot be chosen independently. It is shown below that the effect of expansionary domestic credit creation on the short-run viability of the fixed exchange rate depends on the implications of the policy for the present discounted value of government transfers. If the policy increases present value transfers, then it must cause instantaneous exchange rate collapse. If the policy increases current transfers only, leaving present value transfers unchanged, then it can generate delayed collapse.

A. Initial Equilibrium

To determine the criteria necessary for delayed predictable collapse, it is first necessary to establish an initial equilibrium in which the fixed rate is viable indefinitely. From this initial equilibrium, policy change and its implications for the survival of the fixed rate regime can be analyzed.

This section defines initial values and the corresponding values of the policy variables, such that fixed and flexible rate equilibria are both viable. Assume that ${\mu }_{\mathit{0}}={\mu }^{*}={\dot{F}}_0=0$, such that transfers are constant at the value given by τ₀ = θ F₀ Setting the rate of change of the exchange rate equal to zero in equation (7), and solving yields an expression for the equilibrium fixed exchange rate:

where an overbar is used to denote the value of the exchange rate in the fixed rate regime. The fixed exchange rate is assumed to be set at this value.

To assure that this fixed rate equilibrium is viable, it is necessary to solve for the value the exchange rate would take on, conditional on speculative attack and a switch to a floating exchange rate regime. Flood and Garber (1984) label this the shadow floating exchange rate. Viability requires the absence of speculative profits from attack and therefore requires that the shadow floating rate never exceed the fixed rate.

To solve for the shadow floating rate, it is necessary to specify a reserve commitment by the central bank. Assume that the central bank is willing to lose a maximum of $\overline{S}{\delta }_t$ foreign exchange reserves at time t. Once this quantity is gone, they abandon the fixed rate. Additionally, assume that the impact of the foreign exchange reserve loss on the money supply is not sterilized.⁶ Therefore, with exchange rate collapse, the government loses $\overline{S}{\delta }_0$ foreign exchange reserves, and the money supply decreases by $\overline{S}{\delta }_0$. Domestic agents increase their holdings of real bonds by δ₀ as they swap real bonds for real money balances. The shadow floating rate values of stock variables are denoted by tildes (^∼). At time 0, they are given by:

The shadow flexible exchange rate is that rate which sets $\frac{\stackrel{·}{S}}{S}=\tilde{\mu }$, where $\tilde{\mu }$ is given by the value of μ which sets $\dot{F}=0$ in equation (4), allowing the discrete adjustment of asset stocks through the foreign exchange market. Its value is given by:

Substituting$\tilde{\mu }$ into equation (7), it can be shown that ${\tilde{S}}_0=\overline{S}$. Additionally, since foreign exchange reserves are not changing in the equilibrium defined above, the shadow floating rate in the future continues to equal the spot rate.

Since the shadow floating rate equals the spot rate now and in the future, profits from a speculative attack are zero, and the flexible rate equilibrium is viable indefinitely. Note, however, that if agents do attack and the government loses some foreign exchange reserves, then money growth must be higher to replace the government’s lost interest revenues on foreign exchange reserves, yielding a self-fulfilling exchange rate crisis.⁷ This is a problem of multiple equilibria and is considered at the end of the paper. For now, assume that speculative attack does not occur, and focus on the fixed rate equilibrium.

B. Domestic Credit Expansion and Instantaneous Collapse

From the initial equilibrium, described above, in which there is no expectation of exchange rate collapse, consider a policy change leading to an increase in the rate of domestic credit creation. Specifically, assume that at time 0, the domestic government permanently increases transfers to agents, such that τ = τ₀ + Δ τ = θ F₀ + Δ τ, and finances this with an increase in domestic credit creation according to equation (3).

Consider, first, money market equilibrium under fixed exchange rates. From equation (8):

The increase in transfers increases the present discounted value of disposable income, thereby increasing real money demand. If the exchange rate is to remain fixed, agents must make a discrete swap of real bonds for money, as the authorities peg the exchange rate. The increase in the value of transfer payments creates declining foreign exchange reserves from equation (4), as in Krugman (1979) and Flood and Garber (1984). However, for this fixed rate to be a viable equilibrium, it must be the case that the shadow floating rate not exceed the value of the fixed rate. Otherwise, agents will attack forcing abandonment of the fixed rate.

Now, compute the shadow flexible exchange rate. Under floating rates $\frac{\dot{S}}{S}=\mu$, such that from equation (4), ${\tilde{\mu }}_0{\tilde{m}}_0={\Delta }\tau +\theta {\delta }_0$. After substitution and manipulation of the exchange rate equation, the shadow floating rate relative to the fixed rate can be expressed as:

for reasonable magnitudes of δ.⁸ Therefore, the shadow floating rate is above the fixed rate irrespective of the magnitude of reserve loss. Speculators will attack the fixed rate regime, the instant of the policy change, to reap the implied profits.

It is useful to think about the intuition behind this result. Return to the fundamental equation given by (7). The policy decision is to increase transfer payments to the public and finance this with seigniorage revenue, created by a corresponding rise in the rate of domestic credit creation. This policy change increases the present-discounted value of disposable income, increasing the representative agent’s perception of his wealth. The increase in wealth raises an agent’s desired current and future consumption.⁹ But present-value consumption cannot increase since the present value of consumption resources is unchanged. The attempt to increase current consumption must force a price level increase in equilibrium. This reduces the real value of his nominal assets such that he is content with the initial values for current and future consumption.¹⁰ With the foreign price level fixed, the only way the domestic price level can rise is with exchange rate collapse.

To understand the result in a different way, use equation (4) to substitute for the flexible rate value of $\left({\theta }+\frac{\dot{S}}{S}\right)\frac{M}{S}$ to yield:

Note that the two groups of financial assets $\left[\left(F-\frac{M}{S}\right);\left(\frac{M}{S}+b\right)\right]$ are unaffected by the magnitude of reserve loss. Expenditure on real money balances, given by the left-hand side, is not affected by reserve loss because any reduction in reserves feeds into larger money growth necessary to achieve the target seigniorage revenues. Domestic real wealth is not affected by an exchange of equal values. Therefore, if the equation holds initially with Δ τ = 0, then it cannot hold with Δ τ > 0 at an unchanged exchange rate. From equation (7), the exchange rate must increase.

In summary, when a policy change toward expansionary domestic credit creation increases the present value of government transfers, exchange rate collapse is instantaneous. This suggests that a criteria for the short run viability of fixed exchange rate is that the present value of government transfers not rise. The next section demonstrates the validity of this hypothesis.

C. Domestic Credit Expansion and Delayed Collapse

Now, consider the alternative policy of setting $\dot{D}=\rho D$ and letting τ be determined by equation (4). In the initial equilibrium, $\rho =\dot{D}=0$, and the initial fixed exchange rate is given by equation (8). Note that under this policy, the present discounted value of transfers is conditional on the exchange rate regime. Therefore, to compute the shadow floating exchange rate, it is first necessary to compute the present value of transfers, conditional on exchange rate collapse.

From equation (3), the present value of transfers, conditional on collapse at time zero, becomes:

where $\mu =\rho =\frac{\dot{S}}{S}$ such that:

and

Substituting and integrating yields:

This simplifies further to ${\tilde{\psi }}_0=\frac{p{D}_0}{\theta {\tilde{S}}_0}$, under the conventional assumption that the reserve floor is zero, such that the authorities abandon the fixed rate when reserves are exhausted (F₀ = δ₀). Assuming a zero reserve floor and using equation (7) to solve for the shadow floating exchange rate at time zero yields:

The shadow floating rate can be below, equal to, or above the fixed rate, as in Flood and Garber (1984). Additionally, higher reserves imply a lower shadow floating rate, suggesting that a government can maintain a fixed rate as long as it has sufficient reserves.

To understand why this result differs from the previous one in which expansionary domestic policy required instantaneous exchange rate collapse, independent of the level of reserves, it is necessary to consider the effect of this policy on the present value of transfers. When the policy variable is the rate of expansion of domestic credit, with the transfers determined endogenously from the government’s budget constraint, the present value of transfers does not necessarily increase.¹¹ The relationship of the shadow floating rate to the fixed rate is determined by the relationship of the present value of transfers, conditional on collapse at time zero and evaluated at the fixed exchange rate $\left(\frac{\rho D_0}{\theta \overline{S}}\right)$, to the initial value of transfers, given by F₀ If the present value of transfers, conditional on collapse, does not rise, the shadow floating rate is below the spot rate, and delayed collapse is possible. Collapse today would imply the loss of so many reserves relative to the increase in the present value of seigniorage that the present value of transfers would actually fall. A postponed collapse allows a smaller reserve loss on the day of collapse and higher present value seigniorage revenue, reducing the fall in the present value of transfers.

When delayed collapse is possible, its timing is computed to equate the shadow floating rate at the time of collapse (T) with the fixed rate, as in Flood and Garber (1984). Therefore, collapse occurs when:

where from equations (3) and (4), $F_t=F_0+\frac{D_0}{\overline{S}}\left(1-e^{\rho t}\right)$ for t ≤ T. Substituting, yields an expression which determines the date of collapse (T) as:

As in other work, the time to collapse is increasing in reserves,¹² and decreasing in the rate of growth of domestic credit (ρ).

Delayed collapse is possible only when the increase in the rate of growth of domestic credit creation leaves the present value of transfers, conditional on collapse at time T, unaffected. To see this, compute the present value of transfers at time 0, conditional on collapse at time T as:

Substituting for e^ρT from equation (9), yields: ψ_0,T = F₀. The present value of transfers is constant at its initial level of F₀ , conditional on collapse at time T.

Therefore, for a policy of expansionary domestic credit creation to be consistent with a fixed exchange rate, even temporarily, the policy change must leave the fiscal constraint satisfied. In particular, the present value of government transfers must not deviate from its initial equilibrium level. Otherwise, when agents perceive their wealth has increased, the attempt to increase consumption, with the intention of increasing present-value consumption, when the present value of resources is unchanged, must depreciate the exchange rate immediately, reducing the real value of domestic monetary assets. The level of reserves matters only if it plays a role in determining whether the policy change increases the present value of transfers.¹³

Using the government’s intertemporal budget constraint, which constrains the present value of transfers to equal the present value of seigniorage revenues plus initial debt, it is possible to provide a more general interpretation of the fiscal constraint.¹⁴ Note that an increase in the present value of transfers implies an increase in the present value of seigniorage revenues.¹⁵ Therefore, avoidance of instantaneous collapse constrains the government from increasing the present value of seigniorage revenues.

D. Sterilization and Exchange Rate Collapse

Finally, it is interesting to consider the implications of sterilizing the effect of reserve changes on the money supply. Flood, Garber, and Kramer (1996) and Flood and Marion (1996) claim that it is important to recognize that governments do sterilize. To allow sterilization, it is necessary to introduce a government bond, held by the public, into the model. Assume that the government issues real bonds b^g to private agents and that these are perfect substitutes for foreign real bonds and pay the same interest rate. Any reduction (increase) in the monetary base due to a loss (gain) of foreign exchange reserves is instantaneously offset by an open market purchase (sale) of government bonds to keep the money supply constant. Government non-monetary debt (b^g – F) is constant under a policy of sterilization. It is well known that, with complete sterilization, if the exchange rate is not viable in the long run, then it is not viable in the short-run either. Expansionary domestic credit creation must cause immediate exchange rate collapse, independent of fiscal policy.¹⁶

However, it is also possible to show that sterilization can eliminate the possibility of multiple equilibria for the case where the fixed rate is viable in the long-run. Consider the case where the policy choice variable is the level of transfers. Here, multiple equilibria can arise because a speculative attack reduces the government’s interest-bearing reserves. To maintain transfers at their target level, domestic credit creation and money growth must increase. A policy of sterilizing the effects of these speculative attacks on the money supply would replace foreign exchange reserves with reduced government debt, eliminating the need for higher monetary growth and eliminating self-fulfilling crises. To summarize, if the fixed rate is viable in the long run, and the authorities sterilize the results of speculative attacks, then the fixed rate is viable in the short run. Sterilization assures that multiple equilibria vanish.