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Speculative attacks: The roles of intertemporal substitution and the interest elasticity of the demand for money
KENT P. KIMBROUGH Duke Durham,
Uniuersity
North
Carolina
Speculative Attacks: rite Roles of Intettemporal Substitution and the Interest Elasticity of the Demand for Money* The effects of an anticipated speculative attack and exchange rate regime collapse brought on by an unsustainable mix of domestic credit and exchange rate policies is examined. A maximizing model with money demand motivated by a transactions technology which implies that increased money holdings reduce transactions costs associated with consumption good purchases is used. It is demonstrated that the effects of an impending speculative attack depend crucially on two margins through which the forward-looking behavior of rational consumers manifests itself: the intertemporaf elasticity of substitution in consumption and the interest elasticity of the demand for money.
1. Introduction A notable feature of pegged and crawling peg exchange rate systems is that often the rate at which the exchange rate is pegged or the rate at which the exchange rate is crawling must be abandoned. Typically the cause of abandonment is excessive domestic credit creation in order to meet the government’s fiscal needs. The result of an exchange rate policy that is unsustainable in the face of such domestic credit expansion is a chronic balance of payments deficit, dwindling international reserves, and, ultimately, a collapse of the exchange rate policy marked by a speculative attack on the Central Banks remaining stock of international reserves. Numerous models aimed at describing the effects and timing of speculative attacks when they are anticipated by rational, forward-looking agents have been proposed in recent years. One line of research uses postulated money demand functions combined with rational expectations to model the forward-looking behavior that ap*I would like to thank Phil Brock, Grant Gardner, and an anonymous for helpful comments. Funding for this research was provided in part by from the Sloan Foundation to the Department of Economics.
Journal of Macroeconomics, Fall 1992, Vol. 14, Copyright 0 1992 by Louisiana State University 0164-0704/92/$1.50
No. 4, pp. Press
689-710
referee a grant
689
Kent P. Kimbrough pears to be central to understanding the characteristics of speculative attacks. Examples include the theoretical models of Flood and Garber (1984) and the theoretical and empirical work of Blanc0 and Garber (1986). The key behavioral parameter in these models is the interest sensitivity of the demand for money as it provides the link between anticipated future policies after the collapse and reserve losses under the initial, unsustainable policy prior to the collapse. A second, and more recent, line of research derives the demand for money, along with the equilibrium time-profile for consumption, from the optimizing behavior of consumers. Calvo (1986, 1987) uses a continuous-time cash-in-advance constraint to motivate the demand for money and study the behavior of an economy facing an anticipated speculative attack. As is well known, use of a cashin-advance constraint results in a constant velocity of money. This eliminates the traditional channel for an interest-sensitive money demand function. However, Calvo is able to maintain the forwardlooking behavior that is essential to understanding the impact of a collapse of the exchange rate system through two features of his model. First, since the maximizing behavior of consumers in an intertemporal setting is explicitly accounted for, the equilibrium consumption path is a function of permanent income, or wealth, which is a forward-looking variable. Second, the use of the cash-in-advance constraint in continuous time results in a situation where the opportunity cost of holding money influences consumption decisions. These two features of Calvo’s model, combined with the fact that the demand for money is proportional to consumption, mean that even though the velocity of money is interest inelastic, in equilibrium the demand for money exhibits forward-looking behavior. This allows Calvo to analyze the real and monetary implications of an impending collapse. However, the margin along which anticipations of the collapse matter reflect intertemporal substitution in consumption since it is only through scale effects that the demand for money, in equilibrium, responds to changes in its opportunity cost. Another approach to studying speculative attacks based on optimizing behavior has been to motivate the demand for money by entering money directly into the utility function of the representative consumer. Obstfeld (1986) pursues this approach using a utility function that is separable in consumption and real balances. This specification gives rise to an equilibrium time profile for consumption that is independent of monetary considerations. Even so, since consumers equate the marginal rate of substitution between money 690
Speculative
Attacks
and consumption at each moment in time to the opportunity cost of holding money, the implied demand for money function is sensitive to the interest rate. This allows for the forward-looking behavior required to study the economic effects of a speculative attack. There are, however, no real effects of the impending attack and collapse of the exchange rate policy. Claessens (1988) drops the separability assumption imposed by Obstfeld and is thus able to model the potential real effects associated with an anticipated collapse. However, Claessens’s analysis is restricted to the case of CobbDouglas preferences. The aim of this paper is to allow both intertemporal substitution and the interest elasticity of demand for money to come into play when discussing the effects of a speculative attack and subsequent collapse of a fixed exchange rate regime. Earlier authors have constrained the role of the interest elasticity by using special models of money demand, a separable utility function, or CobbDouglas preferences. The formulation used here is more general. The use of money is motivated by a transactions technology similar to that adopted by Kimbrough (1986). It is shown that, as a first approximation the impact of a speculative attack and anticipated collapse of an unsustainable fixed exchange rate regime depend crucially on whether the intertemporal elasticity of substitution in consumption is greater or less than the interest elasticity of the demand for money. The impact effects, transition dynamics, and the timing of a speculative attack that leads to a collapse of the exchange rate regime are discussed for both cases.
2. The Model
and Benchmark
Equilibrium
The economy under consideration is a small open economy inhabited by a representative consumer with perfect foresight. The consumer’s objective is to maximize lifetime utility which depends on the discounted value of the instantaneous utility from consumption of a single perishable consumption good as given by Jt U(c,)e-“dt
,
where c, is consumption at time t and 6 > 0 is the consumer’s subjective rate of time preference. For concreteness, the instantaneous utility function will be assumed to be characterized by a constant inter-temporal elasticity of substitution in consumption l/y where y > 0 and V(c,) = ciWy/(l - y), y # 1 or U(c,) = In c,, y = 1 . 691
Kent P. Kimbrough The consumption good is freely traded in world goods markets and its price in terms of foreign currency is, for simplicity, taken to be fixed at unity. Goods market arbitrage thus equates the domestic currency price of the consumption good with the exchange rate, S,. Since there is only one good, this implies that the domestic rate of inflation, nr+, equals the rate of depreciation of the domestic currency. Output of the consumption good in the home country is constant at y. There are two assets available to domestic consumers-an internationally traded bond offering a real return of r* that is determined in world markets, and domestic money which is held only by domestic residents. Given the assumptions about goods prices, the domestic nominal interest rate is r* + a,. In view of these features of world goods and asset markets, the consumer’s problem amounts to choosing time profiles for consumption, money, and bonds so as to maximize (1) subject to the lifetime budget constraint
0
I
{[l + u(m,/c,)]c, + (r* + 7r,)m,}e+‘dt
0 m
qe+‘dt + w + b. , (2) 0 where T, is a lump-sum transfer payment from the government, and m,, and b. are the consumer’s initial holdings of money and bonds. The function u(e) characterizes transactions costs associated with consumption. It is assumed that holding money reduces transactions returns to costs (that is, u’ < 0) and that there are diminishing holding money (that is, u” > 0). The transactions cost function is assumed to be given by u(tnJc,) = (+/rt)(mJ~J’-~, q > 1 and + > 0 which satisfies the desired curvature properties and assures u(n) > 0.’ =
y/r*
+
I
‘From Feenstra (1986), the transactions technology used here is equivalent to some utility function with money. To obtain this utility function note from Feenstra’s (14) and (15) that x, e c, + $(ct, m,) and V(x,, mJ = U(G) where x, is “gross consumption.” Since in the current setup +(ct, tnJ = kc#-l, k = 4/q, it follows that x, = c, + kcti-‘. Solving this for c, and substituting into the identity for V(e) yields the utility function with money implied by the transactions technology. For example, for rt = 2, c, = (rn,/2kx[l + (4kx,/m,)]“* - 1). It is evident from this that the implied utility function is neither separable or Cobb-Douglas, as in earlier papers.
692
Speculative Note from the consumer’s optimization holdings satisfy the first-order condition
-v’(m,/c,)
problem
Attacks
that money
= r* + 7r, .
This means that, along an optimal given by
(3)
path, the demand
m, = k(r* + P,)c, ,
k’ = -l/v”
for money is
< 0,
(4)
which, given the functional form of the transactions cost function, implies m, = [q(r* + n,)/(q - l)+]- (lh)c,. The interest elasticity of demand for money is thus l/q. Since q > 1, the model implies this elasticity is less than one. Given the optimal time profile for money holdings as described by (4), the consumer’s problem is to choose the time profile for consumption that maximizes (1) subject to the budget constraint 5
cc
q(r* + n,)c,e+‘dt
= y/r*
+
T,emetdt + m,, + b, , I0
I0
(5)
where q(r* + TJ = 1 + v[k(r*
+ ITJ]
+ (r* + n,)k(r* + IT,) ,
9’ = k(s) > 0
63)
is the effective price of the consumption good in period t. This price includes not only the direct price of the consumption good but also the transactions costs associated with purchasing consumption goods and the opportunity cost of holding money to facilitate their purchase. In addition to the budget constraint (S), the optimal consumption profile satisfies the first-order condition U’(c,)e-” = hq(r* + 7r,)e+’ for all t, where A is the marginal utility of wealth. Assuming that the world real interest rate and the consumer’s rate of time preference are equal (that is, r* = 6), the optimal consumption profile satisfies the condition U’ (c,) = Xq(P + 7TJ 3 c* = [hg(r* To proceed,
further
government
policy
+ P,)]-“y
.
(7)
needs to be charac693
Kent P. Kimbrough terized. It is assumed throughout the analysis that government spending is constant at zero and that no distorting taxes other than the inflation tax are levied. The government’s budget constraint is thus m m
TTteertdt = b$ + (ti, + T,m,)e-“‘dt , 03) I0 I0 where bg is the government’s bond holdings, which shall be viewed as the Central Bank’s international reserves, and riz, + rim, is the revenue from money creation. 2 In order to allow for the possibility of a collapse of the exchange rate regime, the standard assumption of the literature on speculative attacks is adopted, and it is assumed that there is a lower bound on international reserves of zero (that is, bf 2 O).3 The initial government policy is assumed to be sustainable in the sense that it consists of a combination of domestic credit policy and exchange rate policy that can be maintained indefinitely without any need to adjust either policy. In particular, the exchange rate is assumed to be fixed at the level 3 and nominal domestic credit, D,, to be growing at the constant rate lo so that D, = DoePf .
(9)
With this government policy mix the equilibrium time profiles for consumption and real balances will be flat. Since the Central Banks balance sheet implies that the money supply, in real terms, is m, = (DJS) + bf ,
WV
this means that for the fixed exchange rate 3 to be sustainable requires that u. = 0. It is assumed that the initial policy satisfies this condition.
‘More formally, the right-hand side of (8) should also allow for instantaneous jumps in money stock. 30bstfeld (1966) discusses this issue. He shows that there is no lower bound on international reserves. For a policy to be sustainable requires that the rate of growth of domestic credit be less than the real interest rate. This assures that the government budget constraint is consistent with the government paying off its debt in the long run. The unsustainable policy of Section 4 thus might be viewed as arising hecause the upper bound on domestic credit growth is being exceeded.
694
Speculative
Attacks
In addition, the private sector budget constraint (5) and the government budget constraint (8) together imply that the economy’s overall resource constraint is m
(1 + v[k(r* I
+ T,)]}c,ewF’dt = (y/r*)
+ fo ,
(11)
0
where ft = b, + J$ is the economy’s total holdings of foreign bonds. Since under the sustainable policy described above c, = E for all t, it is easy to show using (11) that
5 = (Y + r*fo)/{l + G4-*)1) > since 7r, = 0 under the sustainable policy. It follows from (4) that under this policy m, = rot, where m = k(r*)E .
3. Unsustainable
Policies and Speculative
02) immediately
(13)
Attacks
Suppose that, instead of following the sustainable exchange rate and domestic credit policies outlined in the previous section, the government announces at time t = 0 that it plans to permanently increase the rate of growth of domestic credit to p > 0 while keeping the exchange rate fixed at S. It is also known that, should the Central Banks stock of international reserves ever be depleted, the government plans to allow the exchange rate to float while maintaining domestic credit growth at p, > 0. In order to understand the effects of the new policy, suppose, for the moment, that the new policy is sustainable. In this case inflation remains at its initial level of zero and the optimal paths for consumption and real balances continue to be given by (12) and (13). However, with real balances constant at fi, the exchange rate fixed at S, and domestic credit growing at the rate or,, it is evident from the Central Banks balance sheet (10) that international reserves will be falling continually at the rate p and will eventually reach their lower bound of zero. Since the policy p > 0 and S, = S for all t implies that international reserves reach their lower bound of zero in finite time, it is unsustainable. Rational consumers will understand that the new policy is unsustainable and will anticipate the impending speculative attack and collapse of the fixed 695
Kent P. Kimbrough exchange rate regime. That is, consumers understand that following the collapse the domestic currency will depreciate at the rate of domestic credit expansion and, therefore, under the new policy, the equilibrium time profile for inflation is understood to be lrr, =
0, P,
tzsT t>T’
where T is the time of the collapse. The actual timing lapse is itself an endogenous variable whose equilibrium be determined shortly.
(14) of the colvalue will
Transition and Post-Collapse Periods As a consequence of anticipating the future speculative attack and exchange rate collapse, consumers adjust their consumption and money holdings immediately upon announcement of the new policy rather than waiting until the attack and collapse actually occur. It is clear that consumption will be higher during the transition period prior to the speculative attack than it will be after the attack. To see this, note that the equilibrium level of consumption depends on permanent income and on the time profile of the effective price of consumption. Since the effects of the new policy are anticipated at time t = 0 when it is announced, permanent income jumps discretely at this time and remains constant thereafter. In particular, permanent income does not change at the time of the speculative attack as the impact of the collapse is already accounted for by rational, forward-looking consumers at the time of the announcement of the new policy. However, it can be seen from (6) and (14) that the effective price of consumption increases permanently at the time of the speculative attack that brings down the exchange rate regime. This means that under the new policy there is an intertemporal substitution effect towards consumption during the transition period so that consumption prior to the collapse exceeds consumption after the collapse. In addition, since the opportunity cost of holding money rises permanently at the time of the collapse, money holdings also fall when the collapse occurs. As can be seen from (9, this drop reflects not only a direct effect from the rise in the opportunity cost of holding money, but an indirect effect as well stemming from the scale effects due to the decline in consumption at the time of the collapse. Further, the levels of consumption during the transition period 0 4 t I T and following the collapse are constant (although 696
Speculative
Attacks
at different levels). This follows from the fact that permanent income is constant following the policy announcement and the effective price of consumption, although differing prior to and after the collapse, is constant within each subperiod. Similarly, the demand for money is constant during the subperiods 0 5 t 5 T and t > T. In order to further understand the effects of the new policy and the anticipated speculative attack, it is useful to formally derive the results described above. These results imply that during the transition period 0 I t I T consumption is constant at some level E, while, following the attack, consumption is constant at some level 2, where, from the first-order condition for consumption (7), 2/z = [q(‘*)/q(r*
+ p)p
< 1.
(15)
Similarly, using the same notation to differentiate between the transition period and the post-collapse period, it can be seen from (4) that 732/t% = [k(r* + k)/k(T*)](Z/E) < e/E < 1, indicating that, at the time of the collapse, real balances fall proportionately more than does consumption. Using (15) in the resource constraint (ll), which under the new policy becomes T
I
{ 1 + v[ k(r*)]}c,e-rl’dt
0
+
-{l + v[k(r* IT
it can be demonstrated
ct =
+ p)]}c,e-F’dt
= (y/r*)
that
~5= tj-“‘{(y + r*fo)/[(l e = Q-1/y {(y + r*f,)/[(l
- e-“r)8 - e-“‘)6
+ LeT6]}, + e+T6]} ,
+ ff ,
tt
5 T > T’
(16)
where 9 = q(r*), 9 = q(r* + p), 6 = (1 + v[k(r*)]}g-“‘, and 6 = (1 + v[k(r* + ~)]}4-“‘. For the discussion that follows, it is useful to think of the demand for consumption at time as a function of the effective price of consumption 9t and the marginal utility of wealth A. These “X constant” demand curves are given by [A9(*)],-“’ as can be seen from (7). As discussed by MaCurdy (1981), the y constant demand curve is the natural extension of the permanent income theory of consumption to a situation where the price of consumption varies over time (as it does here because of the impending speculative
t
697
Kent P. Kinabrough attack). In such settings the marginal utility of wealth, like permanent income, as it is usually defined, summarizes all information about the future path of the economy that is relevant for current consumption choices. Only when the effective price of consumption is constant is the standard concept of permanent income adequate. Applying the A constant demand concept, it can be seen from (15) and (16) that the q-‘lr terms in (16) capture the substitution effects of the new policy. It is also apparent that the term in braces in (16), which equals A-“‘, captures the influence of shifts in the marginal utility of wealth on the height of the equilibrium consumption profile. Just like the standard notion of permanent income, this term determines the height of the consumption profile, and from here on (y + r*fo)/[(l shall be referred
Announcement
- e+‘)4
to as permanent
+ e+%]
income.
Effects
One issue that has not been dealt with so far, but that is an important feature of an anticipated speculative attack, is the shift in macroeconomic variables at the time the new policy that precipitates the collapse is announced. At the time the new policy is announced, the effective price of consumption remains at its initial level, 0 = q(e). Th e announcement effect of the new policy on the equilibrium consumption profile is thus a pure wealth effect capturing the shift in permanent income that is associated with the announcement. Consumption rises at the time the unsustainable policy is announced if permanent income rises, and falls if permanent income falls. Additionally, since the opportunity cost of holding money is unchanged at the time of the policy announcement, the behavior of real balances is proportional to the change in consumption. Note from (12) that consumption under the original, sustainable policy can be written as Q-l’v times permanent income under the original policy, which is given by (y + r*fo/[(l
- e-“‘)I,
+ e+‘h]
.
08)
As can be seen from (17) and (18), permanent income rises or falls with the policy announcement as B S 8. From the preceding discussion it follows that
696
Speculative
Attacks
and
as
esii,
(19)
where
6(r* + a,) s (1 + v[k(r* + IT,)]}~-I”. Two factors thus work to influence permanent income at the time of the policy announcement. On the one hand, the opportunity cost of holding money increases after the collapse, reducing the money-consumption ratio (the inverse of the velocity of money) and thereby raising per unit transactions costs during the post-collapse period (that is, v[k(r* + TV)] > v[k(r*)]). This works to lower permanent income at the time of the policy announcement and tends to reduce consumption and real balances. The magnitude of this effect depends crucially on the interest elasticity of demand for money l/q. On the other hand, after the collapse the effective price of consumption is higher (4 > 0). This discourages consumption after the collapse and serves to reduce the resources devoted to transacting after the collapse. As a result, permanent income tends to increase at the time of the policy announcement raising consumption and real balances. The strength of this effect depends on the intertemporal elasticity of substitution in consumption l/-y.4 The preceding discussion suggests that whether or not permanent income rises or falls at the time of the policy announcement depends on the relative magnitudes of the inter-temporal elasticity of substitution in consumption and the interest elasticity of the demand for money. To see the exact nature of the dependence of the shift in permanent income at the time of the announcement on these two elasticities, recall (19) and note that 6 Z 6 as 8’ 2 0. From the definition of 0(m) it can be seen that 8’ = q-l’Y[v’k’ - (1 + v)(k/ y9)]. Using the fact that k’ = -(l/v”) along with the functional form
Q is important to note that both factors influencing permanent income at the time of the policy announcement depend on transactions costs. As a result, the possibility that permanent income may rise or fall with the policy announcement does not depend in any fkdamental sense on whether or not transactions costs are “large. ”
699
Kent P. Kirnbrough for u(e) and Equations (4), (6), and (12), it can be shown that 0’ = l~q-“~[(l/q) - (l/y)(l + &‘I, where p = r*k~/(y + r*fo is the permanent share of seigniorage in national income under the sustainable policy. This, in conjunction with (19) implies that
as WY)(l
+ PY’ 3 l/q
.
(20)
As will be discussed later, the share of seigniorage in national income is typically quite small so that, roughly speaking, (20) indicates that the impact of the new policy at the time it is announced depends on whether or not the intertemporal elasticity of substitution in consumption is greater or less than the interest elasticity of the demand for money. If the intertemporal elasticity of substitution exceeds the interest elasticity of the demand for money, permanent income rises when the unsustainable policy is announced and consumption and real balances also increase. If the reverse is true then the announcement effect of the unsustainable policy is to reduce permanent income and lower consumption and real balantes . In contrast to the announcement effects of the unsustainable policy, and as can be verified from (12) and (16), the post-collapse level of consumption is always less than the level of consumption under the sustainable policy regardless of the relative magnitudes of the intertemporal elasticity of substitution in consumption and the interest elasticity of the demand for money. This can be seen by noting that 6 can be written as y + r*fO times the inverse of the expression in square brackets in (21). Since the term in square brackets is a weighted average of two terms greater than 1 + 6, 2 < E. Intuitively, the inter-temporal substitution effects of the policy unambiguously outweigh the wealth effects. Since the opportunity cost of holding money also rises at the time of the collapse, the post-collapse level of real balances must also be lower than the level of real balances under the sustainable policy. The upper panels of Figures 1 and 2 illustrate the two possible time profiles for con700
Speculative
Attacks
bg0
T
t
bg0
-Q-
WYU
Figure 1. + PY > I/?
sumption and real balances following tainable policy.
the enactment
of the unsus-
The Balance of Payments Accounts Another important issue relating to the effects of unsustainable policies and anticipated speculative attacks concerns the behavior of the various balance of payments accounts. To begin, note from (20) that, as a first approximation, if the intertemporal elasticity of substitution in consumption is greater than the interest elasticity of the demand for money, real balances rise at the time of the policy announcement while if the reverse is true real balances fall. If discrete jumps in domestic credit do not occur, the Central Banks stock of international reserves may increase or decrease when the 701
Kent P. Kimbrough m.
lil ii
m t
(l/Y)(l
Figure 2. + PY <
‘i I I I I I I
.
;
I I
b
T
t
WI
unsustainable policy is announced. In fact, if the intertemporal elasticity of substitution in consumption is less than the interest elasticity of the demand for money, the unsustainable policy will be characterized by two attacks on the Central Banks stock of international reserves: the first attack occurs at the time the unsustainable policy is announced, and the second attack results in the collapse of the fixed exchange rate. In practice, these jumps in international reserves are likely to be spread out over a short period of time following the announcement of the new policy. However, after this brief adjustment period, the balance of payments will exhibit a chronic deficit up until the time of the speculative attack and the collapse of the exchange rate regime. These results are 702
Speculative
Attacks
summarized by the time profiles for international reserves shown in Figures 1 and 2. It can also be demonstrated that the behavior of the current account depends on the intertemporal elasticity of substitution in consumption and the interest elasticity of the demand for money. The current account is defined by ca, = y + r*ft - (1 + v[k(r* + ~FJ]}. From this and the fact that the consumption profile under the sustainable policy is flat, it follows that prior to the policy announcement the current account is balanced in all periods. At the instant the new policy is announced the current account is y + r*fO - (1 + v[k(r*)]}E. It follows from (20) that the current account deteriorates or improves at the time of the policy announceif the intertemporal elasment as (l/y)(l + p)-’ 2 l/q. Intuitively, ticity of substitution in consumption is large enough, permanent income rises with the policy announcement and the current account deteriorates. The decline in the economy’s stock of foreign bonds causes the current account to continue to deteriorate throughout the entire transition period 0 5 t 5 T. When the collapse occurs, the current account immediately becomes balanced and remains balanced thereafter. Alternatively, with a sufficiently low intertemporal elasticity of substitution in consumption, the policy announcement lowers permanent income and improves the current account. In this case the current account steadily improves until it once again becomes balanced at the time of the collapse. Figures 1 and 2 illustrate the two alternative paths for the current account. One implication of these results is that the current account is not a reliable indicator of an unsustainable mix of domestic credit and exchange rate policies. Under certain conditions the current account may actually improve when an unsustainable policy is enacted and continue to improve until the collapse takes place. Looking at things through the current account window may thus fail to signal the impending speculative attack and collapse. However, the chronic loss of international reserves through the balance of payments will accurately reflect the unsustainable nature of the policy. Speculative Attacks and Utility It is evident that, in the case illustrated by Figure 2 where the intertemporal elasticity of substitution is low, and hence (l/y)(l + p)-’ > I/q, utility falls upon anticipation of the speculative attack because consumption declines in all periods relative to what it would have been under the sustainable policy. However, when the intertemporal elasticity of substitution is high, it is not readily apparent 703
Kent P. Kimbrough that utility is reduced by the unsustainable policy since consumption is actually higher up until the time of the collapse. To formally derive the effects on utility of the increased rate of domestic credit growth underlying the anticipated speculative attack, note from (1) that in the presence of an anticipated speculative attack U = (1 - e+r)[C’-Y/r*(l
- r)] + e+T[2’-Y/P(l
- y)] .
Taking the derivative of this expression with respect to the rate of domestic credit growth and evaluating it at the initial equilibrium under the sustainable policy (where E = 2 so that the XT/+ term drops out of the resulting expression) yields dU/+
= (P/r*)[(l
- e+r)(fZ/a~)
+ e+‘(S/f!+.)]
.
The change in utility is the marginal utility of consumption, P, times the present value of the permanent change in consumption or the change in the “average” height of the consumption profile. However, from the budget constraint, T
(1 + 6)Ee-“‘dt
+
-(I + B)2ee-“‘dt = (y/r*) IT
it follows that in the neighborhood 6 = 8 and E = 2).
LO - e+‘)(dE/+J where 6 = u[k(r*)] falls by
NJ/+
+ e+T(dE/dpJ]
+ fo ,
of the initial equilibrium
= -u’k’EeweT/(l
+ i$ < 0 ,
and 8 3 u[k( r* + IL)]. This implies
= -u’k’E1-Ye+T/r*
(where
that utility
(1 + 6) .
This drop in utility, which occurs in all cases, is proportional to the present value of the rise in transactions costs associated with the unsustainable policy triggered by the rise in the rate of domestic credit expansion. The Timing of the Speculutiue Attack All that remains to fully characterize the impact of an anticipated speculative attack is to determine the actual timing of the 704
Speculative
Attacks
attack itself. When the attack occurs the Central Banks remaining stock of international reserves will be depleted and, unable to defend the fixed exchange rate S, the government will allow the domestic currency to float. Money market equilibrium at the instant of the collapse thus requires that k(r* + h)Z = l&ewT/S,. Since consumers correctly anticipate the collapse, the exchange rate cannot jump at the time of the collapse; if it did, unexploited profit opportunities would exist. Therefore, ST = S must also hold at the time of the collapse. This, (16), and the condition of money market equilibrium imply that the timing of the speculative attack is determined by the condition
k(r* + PXY+ r*fo)l(* - 6) = e@[(l - e-“r)(l
+ S)(o/f!j)“’
+ e+“‘(l
+ S)] ,
(21)
where 6 = v[k(r*)], i7 3 v[k(r* + CL)], and use has been made of the fact that (10) implies that at Do = S(ti - ba. Equation (21) implicitly determines the time of the collapse as a function of the model’s parameters and state variables at the time of the policy announcement. In particular, the time of the collapse depends on the rate of domestic credit creation under the unsustainable policy and on the initial stock of international reserves. For purposes of illustration, consider the effects of changes in the rate of domestic credit creation, CL, on the timing of the speculative attack. As can be seen from (21), given T, an increase in i.~ lowers money demand at the time of the collapse while simultaneously raising the supply of money. The direct effect of an increase in the rate of domestic credit expansion is thus to create an excess supply of money at the time of the collapse. In order to maintain equilibrium, the time at which the collapse occurs must adjust. Changes in T influence both money supply and money demand at the time of the collapse. It is readily apparent that an earlier collapse reduces the supply of money at the time of the collapse thus tending to restore equilibrium in the face of an increased rate of domestic credit creation. If (l/y)(l + p)-’ > l/q, then I + 8 - (1 + fi)((i/~) I” > 0 and an earlier collapse raises permanent income and the demand for money at the time of the collapse (the term in square brackets in [21] falls). This reinforces the equilibrating tendency on the supply-side of the market and hence, when (l/y)( 1 + p)-’ > l/q, the collapse occurs earlier in 705
Kent P. Kimbrough response to an increase in the rate of domestic credit creation (that is, I~T/+L < 0). However, if (l/y)(l + p)-’ < l/q, an earlier collapse lowers permanent income tending to create an even greater excess supply of money at the time of the collapse. If this effect is strong enough to offset the supply-side effect of an earlier collapse, it is possible that the collapse may occur at a later date when the rate of domestic credit creation is increased (that is, aT/ap > 0). Note, however, that the equilibrium time profiles for various macroeconomic aggregates may be as shown in Figure 2, yet an increase in domestic credit growth may nonetheless hasten the collapse. The conditions for a rise in the rate of domestic credit creation to postpone the collapse are more stringent than simply (l/y) (1 + p)-’ < l/q. Similar results can be shown to characterize the impact of the initial level of international reserves on the timing of the collapse. Typically a larger stock of international reserves will postpone the collapse (dT/dbg > 0). However, if the intertemporal elasticity of substitution in consumption is low enough, it is possible that larger international reserve holdings may hasten the collapse (dT/dbg
< 0).
4. Concluding
Remarks
The economic effects of an anticipated speculative attack and exchange rate regime collapse brought on by an unsustainable mix of domestic credit and exchange rate policies have been studied here. The approach has been to use a maximizing model with the demand for money motivated by a transactions technology which implies that increased money holdings reduce the transactions costs associated with consumption goods purchases. It has been demonstrated that the effects of an impending speculative attack depend crucially on two margins through which the forward-looking behavior of rational agents manifests itself: the intertemporal elasticity of substitution in consumption and the interest elasticity of the demand for money. Formally, it has been shown that the impact of a speculative attack depends on whether (mw
+ PY’ 2 l/q
>
where l/y is the intertemporal elasticity of substitution in consumption, I/q is the interest elasticity of the demand for money, 706
Speculative
Attacks
and p is the share of seigniorage in national income. However, evidence presented by Fischer (1982) indicates that seigniorage as a share of national income is typically quite small; usually in the range of I-3% with many countries having smaller shares and very few having shares greater than 4% of national income. Therefore, roughly speaking, the results of the paper indicate that the effects of an anticipated speculative attack depend on whether or not the intertemporal elasticity of substitution in consumption is greater or less than the interest elasticity of the demand for money. If the intertemporal elasticity of substitution in consumption exceeds the interest elasticity of the demand for money, an anticipated speculative attack raises consumption and real balances on impact, and results in an immediate and continued deterioration of the current account until the time of the attack. In contrast, if the intertemporal elasticity of substitution in consumption is less than the interest elasticity of the demand for money, an anticipated speculative attack lowers consumption and real balances on impact, and results in an immediate and continued improvement of the current account until the time of the attack that brings about the collapse of the exchange rate. In addition, in this case there may also be an attack on the Central Banks stock of international reserves at the time the unsustainable policy is announced. It is useful to contrast these results to those of other models of speculative attacks grounded in maximizing behavior. Calvo (1986, 1987) motivates money via a continuous time cash-in-advance constraint. This approach sets the interest elasticity of demand for money to zero. Calvo’s results can thus be viewed as a special case of those obtained here. In essence, his use of the cash-in-advance constraint to motivate money amounts to assuming that the inter-temporal elasticity of substitution in consumption exceeds the interest elasticity of the demand for money. The policy triggering the collapse that Calvo studies is different than the one studied here. He considers a situation where at t = 0 the rate of crawl of the domestic currency (that is, the rate of inflation) is temporarily reduced while domestic credit growth is maintained. Like the policy studied here, it is excessive domestic credit growth that gives rise to the collapse. If Calvo’s policy is examined using the framework of this paper, it is straightforward to show that his results hold only when (l/y) (1 + p)-’ > l/q. Another maximizing approach to studying speculative attacks has been to put money directly into the utility function. Obstfeld (1986) does this under the assumption that the utility function is 707
Kent P. Kimbrough separable in consumption and real balances. As a consequence, unlike the setup used here, Obstfeld’s model does not predict any real effects from an anticipated speculative attack. Claessens (1988) drops the separability assumption and extends Obstfeld’s framework to allow for real effects associated with speculative attacks. However, he does this by assuming that the instantaneous utility function is CobbDouglas, which amounts to constraining the interest elasticity of demand for money to equal one. This perhaps overstates the interest sensitivity of the demand for money. More to the point, as a result of constraining the interest elasticity of the demand for money to equal one, Claessens fails to notice that what is crucial for determining the impact of an anticipated speculative attack is the magnitude of the intertemporal elasticity of substitution in consumption relative to the interest sensitivity of the demand for money. The model presented here can thus be viewed as nesting other maximizing models that have been used to study speculative attacks, In view of this, it is important to know how useful the increased generality of the setup employed here might be in practice. One way to gain perspective on this is to ask what the empirical literature has to say about the model’s two crucial parameters, the intertemporal elasticity of substitution in consumption and the interest elasticity of demand for money. Hansen and Singleton’s (1983) estimates suggest an intertemporal elasticity of substitution in consumption of no less than l/2 and perhaps much greater. However, Hall (1988) has recently presented evidence that leads him to conclude that the intertemporal elasticity of substitution in consumption is “unlikely to be much above 0.1 and may well be zero.” Both of these studies use U.S. data. Regarding the interest elasticity of demand for money, Boorman’s (1972) survey paper suggests that this elasticity ranges from 0.1-0.2 to 0.35-0.75 depending on whether one uses short-term or long-term interest rates. Friedman and Schwartz (1982) estimate the semielasticity of demand for money to be around 10. For nominal interest rates between 5% and lo%, this implies an interest elasticity of demand for money in the range 1/2-l. These studies also use U.S. data. Blejer (1978) and Khan (1977) look at South American countries and report semielasticity estimates that are consistent with an interest elasticity of demand for money in the range of 0.1-1.5 (for nominal interest rates in the range of 20-308, which are reasonable for the period and countries they study). These studies indicate considerable overlap in the range of estimates for the intertemporal elasticity of substitution in consumption on the one hand and the interest elasticity of the demand 708
SpecuZative Attacks for money on the other. In conjunction with the model presented here, this suggests that, in practice, anticipated speculative attacks may not be characterized by a similar response of macroeconomic aggregates such as consumption, real balances, and the current account in all countries at all times. Received: December 1990 Final
version:
January
1992
References Blanco, Herminio, and Speculative
and Peter M. Garber. “Recurrent Devaluation Attacks on the Mexican Peso.” Journal of Political Economy 94 (February 1986): 148-66. Blejer, Mario I. “Black-Market Exchange-Rate Expectations and the Domestic Demand for Money.” JournuZ of Monetary Economics 4 (November 1978): 767-73. Boorman, John T. “The Evidence on the Demand for Money: Theoretical Formulations and Empirical Results.” In Money Supply, Money Demand, and Macroeconomic Models. Edited by J. T. Boorman and T. M. Havrilesky, 248-91. Northbrook, IL: AHM Publishing, 1972. Calvo, Guillermo A. “Temporary Stabilization: Predetermined Exchange Rates.” Journal of PoZiticaZ Economy 94 (December 1986): 1319-29. -. “Balance of Payment Crises in a Cash-in-Advance Economy.” journal of Money, Credit, and Banking 19 (February 1987): 19-32. Claessens, Stijn. “Balance-of-Payments Crises in a Perfect Foresight Optimizing Model.” Journal of Znternational Money and Finance 7 (December 1988): 363-72. Feenstra, Robert C. “Functional Equivalence Between Liquidity Costs and the Utility of Money.” journal of Monetary Economics 17 (March 1986): 271-91. Fischer, Stanley. “Seigniorage and the Case for a National Money.” Journal of Political Economy 96 (April 1982): 295-313. Flood, Robert P., and Peter M. Garber. “Collapsing Exchange-Rate Regimes: Some Linear Examples.” Journal of International Economics 17 (August 1984): 1-13. Friedman, Milton, and Anna J. Schwartz. Monetary Trends in the United States and the United Kingdom. Chicago: University of Chicago Press, 1982. 709
Kent P. Kimbrough
Hall, Robert E. “Intertemporal Substitution in Consumption.” Journal of Political Economy 96 (April 1988): 339-57. Hansen, Lars Peter, and Kenneth J. Singleton. “Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns. ” Journal of Political Economy 91 (April 1983): 249-65. Khan, Mohsin S. “Variable Expectations and the Demand for Money in High Inflation Countries.” The Manchester School of Economic and Social Studies 45 (September 1977): 270-93. Kimbrough, Kent P. “The Optimum Quantity of Money Rule in the Theory of Public Finance.” Journal of Monetary Economics 18 (November 1986): 277-84. MaCurdy, Thomas E. “An Empirical Model of Labor Supply in a Life-Cycle Setting.” Journal of Politica Economy 89 (December 1981): 1059-85. Obstfeld, Maurice. “Speculative Attack and the External Constraint in a Maximizing Model of the Balance of Payments.” Canadian Journal of Economics 19 (February 1986): l-22.
710
Uniuersity
North
Carolina
Speculative Attacks: rite Roles of Intettemporal Substitution and the Interest Elasticity of the Demand for Money* The effects of an anticipated speculative attack and exchange rate regime collapse brought on by an unsustainable mix of domestic credit and exchange rate policies is examined. A maximizing model with money demand motivated by a transactions technology which implies that increased money holdings reduce transactions costs associated with consumption good purchases is used. It is demonstrated that the effects of an impending speculative attack depend crucially on two margins through which the forward-looking behavior of rational consumers manifests itself: the intertemporaf elasticity of substitution in consumption and the interest elasticity of the demand for money.
1. Introduction A notable feature of pegged and crawling peg exchange rate systems is that often the rate at which the exchange rate is pegged or the rate at which the exchange rate is crawling must be abandoned. Typically the cause of abandonment is excessive domestic credit creation in order to meet the government’s fiscal needs. The result of an exchange rate policy that is unsustainable in the face of such domestic credit expansion is a chronic balance of payments deficit, dwindling international reserves, and, ultimately, a collapse of the exchange rate policy marked by a speculative attack on the Central Banks remaining stock of international reserves. Numerous models aimed at describing the effects and timing of speculative attacks when they are anticipated by rational, forward-looking agents have been proposed in recent years. One line of research uses postulated money demand functions combined with rational expectations to model the forward-looking behavior that ap*I would like to thank Phil Brock, Grant Gardner, and an anonymous for helpful comments. Funding for this research was provided in part by from the Sloan Foundation to the Department of Economics.
Journal of Macroeconomics, Fall 1992, Vol. 14, Copyright 0 1992 by Louisiana State University 0164-0704/92/$1.50
No. 4, pp. Press
689-710
referee a grant
689
Kent P. Kimbrough pears to be central to understanding the characteristics of speculative attacks. Examples include the theoretical models of Flood and Garber (1984) and the theoretical and empirical work of Blanc0 and Garber (1986). The key behavioral parameter in these models is the interest sensitivity of the demand for money as it provides the link between anticipated future policies after the collapse and reserve losses under the initial, unsustainable policy prior to the collapse. A second, and more recent, line of research derives the demand for money, along with the equilibrium time-profile for consumption, from the optimizing behavior of consumers. Calvo (1986, 1987) uses a continuous-time cash-in-advance constraint to motivate the demand for money and study the behavior of an economy facing an anticipated speculative attack. As is well known, use of a cashin-advance constraint results in a constant velocity of money. This eliminates the traditional channel for an interest-sensitive money demand function. However, Calvo is able to maintain the forwardlooking behavior that is essential to understanding the impact of a collapse of the exchange rate system through two features of his model. First, since the maximizing behavior of consumers in an intertemporal setting is explicitly accounted for, the equilibrium consumption path is a function of permanent income, or wealth, which is a forward-looking variable. Second, the use of the cash-in-advance constraint in continuous time results in a situation where the opportunity cost of holding money influences consumption decisions. These two features of Calvo’s model, combined with the fact that the demand for money is proportional to consumption, mean that even though the velocity of money is interest inelastic, in equilibrium the demand for money exhibits forward-looking behavior. This allows Calvo to analyze the real and monetary implications of an impending collapse. However, the margin along which anticipations of the collapse matter reflect intertemporal substitution in consumption since it is only through scale effects that the demand for money, in equilibrium, responds to changes in its opportunity cost. Another approach to studying speculative attacks based on optimizing behavior has been to motivate the demand for money by entering money directly into the utility function of the representative consumer. Obstfeld (1986) pursues this approach using a utility function that is separable in consumption and real balances. This specification gives rise to an equilibrium time profile for consumption that is independent of monetary considerations. Even so, since consumers equate the marginal rate of substitution between money 690
Speculative
Attacks
and consumption at each moment in time to the opportunity cost of holding money, the implied demand for money function is sensitive to the interest rate. This allows for the forward-looking behavior required to study the economic effects of a speculative attack. There are, however, no real effects of the impending attack and collapse of the exchange rate policy. Claessens (1988) drops the separability assumption imposed by Obstfeld and is thus able to model the potential real effects associated with an anticipated collapse. However, Claessens’s analysis is restricted to the case of CobbDouglas preferences. The aim of this paper is to allow both intertemporal substitution and the interest elasticity of demand for money to come into play when discussing the effects of a speculative attack and subsequent collapse of a fixed exchange rate regime. Earlier authors have constrained the role of the interest elasticity by using special models of money demand, a separable utility function, or CobbDouglas preferences. The formulation used here is more general. The use of money is motivated by a transactions technology similar to that adopted by Kimbrough (1986). It is shown that, as a first approximation the impact of a speculative attack and anticipated collapse of an unsustainable fixed exchange rate regime depend crucially on whether the intertemporal elasticity of substitution in consumption is greater or less than the interest elasticity of the demand for money. The impact effects, transition dynamics, and the timing of a speculative attack that leads to a collapse of the exchange rate regime are discussed for both cases.
2. The Model
and Benchmark
Equilibrium
The economy under consideration is a small open economy inhabited by a representative consumer with perfect foresight. The consumer’s objective is to maximize lifetime utility which depends on the discounted value of the instantaneous utility from consumption of a single perishable consumption good as given by Jt U(c,)e-“dt
,
where c, is consumption at time t and 6 > 0 is the consumer’s subjective rate of time preference. For concreteness, the instantaneous utility function will be assumed to be characterized by a constant inter-temporal elasticity of substitution in consumption l/y where y > 0 and V(c,) = ciWy/(l - y), y # 1 or U(c,) = In c,, y = 1 . 691
Kent P. Kimbrough The consumption good is freely traded in world goods markets and its price in terms of foreign currency is, for simplicity, taken to be fixed at unity. Goods market arbitrage thus equates the domestic currency price of the consumption good with the exchange rate, S,. Since there is only one good, this implies that the domestic rate of inflation, nr+, equals the rate of depreciation of the domestic currency. Output of the consumption good in the home country is constant at y. There are two assets available to domestic consumers-an internationally traded bond offering a real return of r* that is determined in world markets, and domestic money which is held only by domestic residents. Given the assumptions about goods prices, the domestic nominal interest rate is r* + a,. In view of these features of world goods and asset markets, the consumer’s problem amounts to choosing time profiles for consumption, money, and bonds so as to maximize (1) subject to the lifetime budget constraint
0
I
{[l + u(m,/c,)]c, + (r* + 7r,)m,}e+‘dt
0 m
qe+‘dt + w + b. , (2) 0 where T, is a lump-sum transfer payment from the government, and m,, and b. are the consumer’s initial holdings of money and bonds. The function u(e) characterizes transactions costs associated with consumption. It is assumed that holding money reduces transactions returns to costs (that is, u’ < 0) and that there are diminishing holding money (that is, u” > 0). The transactions cost function is assumed to be given by u(tnJc,) = (+/rt)(mJ~J’-~, q > 1 and + > 0 which satisfies the desired curvature properties and assures u(n) > 0.’ =
y/r*
+
I
‘From Feenstra (1986), the transactions technology used here is equivalent to some utility function with money. To obtain this utility function note from Feenstra’s (14) and (15) that x, e c, + $(ct, m,) and V(x,, mJ = U(G) where x, is “gross consumption.” Since in the current setup +(ct, tnJ = kc#-l, k = 4/q, it follows that x, = c, + kcti-‘. Solving this for c, and substituting into the identity for V(e) yields the utility function with money implied by the transactions technology. For example, for rt = 2, c, = (rn,/2kx[l + (4kx,/m,)]“* - 1). It is evident from this that the implied utility function is neither separable or Cobb-Douglas, as in earlier papers.
692
Speculative Note from the consumer’s optimization holdings satisfy the first-order condition
-v’(m,/c,)
problem
Attacks
that money
= r* + 7r, .
This means that, along an optimal given by
(3)
path, the demand
m, = k(r* + P,)c, ,
k’ = -l/v”
for money is
< 0,
(4)
which, given the functional form of the transactions cost function, implies m, = [q(r* + n,)/(q - l)+]- (lh)c,. The interest elasticity of demand for money is thus l/q. Since q > 1, the model implies this elasticity is less than one. Given the optimal time profile for money holdings as described by (4), the consumer’s problem is to choose the time profile for consumption that maximizes (1) subject to the budget constraint 5
cc
q(r* + n,)c,e+‘dt
= y/r*
+
T,emetdt + m,, + b, , I0
I0
(5)
where q(r* + TJ = 1 + v[k(r*
+ ITJ]
+ (r* + n,)k(r* + IT,) ,
9’ = k(s) > 0
63)
is the effective price of the consumption good in period t. This price includes not only the direct price of the consumption good but also the transactions costs associated with purchasing consumption goods and the opportunity cost of holding money to facilitate their purchase. In addition to the budget constraint (S), the optimal consumption profile satisfies the first-order condition U’(c,)e-” = hq(r* + 7r,)e+’ for all t, where A is the marginal utility of wealth. Assuming that the world real interest rate and the consumer’s rate of time preference are equal (that is, r* = 6), the optimal consumption profile satisfies the condition U’ (c,) = Xq(P + 7TJ 3 c* = [hg(r* To proceed,
further
government
policy
+ P,)]-“y
.
(7)
needs to be charac693
Kent P. Kimbrough terized. It is assumed throughout the analysis that government spending is constant at zero and that no distorting taxes other than the inflation tax are levied. The government’s budget constraint is thus m m
TTteertdt = b$ + (ti, + T,m,)e-“‘dt , 03) I0 I0 where bg is the government’s bond holdings, which shall be viewed as the Central Bank’s international reserves, and riz, + rim, is the revenue from money creation. 2 In order to allow for the possibility of a collapse of the exchange rate regime, the standard assumption of the literature on speculative attacks is adopted, and it is assumed that there is a lower bound on international reserves of zero (that is, bf 2 O).3 The initial government policy is assumed to be sustainable in the sense that it consists of a combination of domestic credit policy and exchange rate policy that can be maintained indefinitely without any need to adjust either policy. In particular, the exchange rate is assumed to be fixed at the level 3 and nominal domestic credit, D,, to be growing at the constant rate lo so that D, = DoePf .
(9)
With this government policy mix the equilibrium time profiles for consumption and real balances will be flat. Since the Central Banks balance sheet implies that the money supply, in real terms, is m, = (DJS) + bf ,
WV
this means that for the fixed exchange rate 3 to be sustainable requires that u. = 0. It is assumed that the initial policy satisfies this condition.
‘More formally, the right-hand side of (8) should also allow for instantaneous jumps in money stock. 30bstfeld (1966) discusses this issue. He shows that there is no lower bound on international reserves. For a policy to be sustainable requires that the rate of growth of domestic credit be less than the real interest rate. This assures that the government budget constraint is consistent with the government paying off its debt in the long run. The unsustainable policy of Section 4 thus might be viewed as arising hecause the upper bound on domestic credit growth is being exceeded.
694
Speculative
Attacks
In addition, the private sector budget constraint (5) and the government budget constraint (8) together imply that the economy’s overall resource constraint is m
(1 + v[k(r* I
+ T,)]}c,ewF’dt = (y/r*)
+ fo ,
(11)
0
where ft = b, + J$ is the economy’s total holdings of foreign bonds. Since under the sustainable policy described above c, = E for all t, it is easy to show using (11) that
5 = (Y + r*fo)/{l + G4-*)1) > since 7r, = 0 under the sustainable policy. It follows from (4) that under this policy m, = rot, where m = k(r*)E .
3. Unsustainable
Policies and Speculative
02) immediately
(13)
Attacks
Suppose that, instead of following the sustainable exchange rate and domestic credit policies outlined in the previous section, the government announces at time t = 0 that it plans to permanently increase the rate of growth of domestic credit to p > 0 while keeping the exchange rate fixed at S. It is also known that, should the Central Banks stock of international reserves ever be depleted, the government plans to allow the exchange rate to float while maintaining domestic credit growth at p, > 0. In order to understand the effects of the new policy, suppose, for the moment, that the new policy is sustainable. In this case inflation remains at its initial level of zero and the optimal paths for consumption and real balances continue to be given by (12) and (13). However, with real balances constant at fi, the exchange rate fixed at S, and domestic credit growing at the rate or,, it is evident from the Central Banks balance sheet (10) that international reserves will be falling continually at the rate p and will eventually reach their lower bound of zero. Since the policy p > 0 and S, = S for all t implies that international reserves reach their lower bound of zero in finite time, it is unsustainable. Rational consumers will understand that the new policy is unsustainable and will anticipate the impending speculative attack and collapse of the fixed 695
Kent P. Kimbrough exchange rate regime. That is, consumers understand that following the collapse the domestic currency will depreciate at the rate of domestic credit expansion and, therefore, under the new policy, the equilibrium time profile for inflation is understood to be lrr, =
0, P,
tzsT t>T’
where T is the time of the collapse. The actual timing lapse is itself an endogenous variable whose equilibrium be determined shortly.
(14) of the colvalue will
Transition and Post-Collapse Periods As a consequence of anticipating the future speculative attack and exchange rate collapse, consumers adjust their consumption and money holdings immediately upon announcement of the new policy rather than waiting until the attack and collapse actually occur. It is clear that consumption will be higher during the transition period prior to the speculative attack than it will be after the attack. To see this, note that the equilibrium level of consumption depends on permanent income and on the time profile of the effective price of consumption. Since the effects of the new policy are anticipated at time t = 0 when it is announced, permanent income jumps discretely at this time and remains constant thereafter. In particular, permanent income does not change at the time of the speculative attack as the impact of the collapse is already accounted for by rational, forward-looking consumers at the time of the announcement of the new policy. However, it can be seen from (6) and (14) that the effective price of consumption increases permanently at the time of the speculative attack that brings down the exchange rate regime. This means that under the new policy there is an intertemporal substitution effect towards consumption during the transition period so that consumption prior to the collapse exceeds consumption after the collapse. In addition, since the opportunity cost of holding money rises permanently at the time of the collapse, money holdings also fall when the collapse occurs. As can be seen from (9, this drop reflects not only a direct effect from the rise in the opportunity cost of holding money, but an indirect effect as well stemming from the scale effects due to the decline in consumption at the time of the collapse. Further, the levels of consumption during the transition period 0 4 t I T and following the collapse are constant (although 696
Speculative
Attacks
at different levels). This follows from the fact that permanent income is constant following the policy announcement and the effective price of consumption, although differing prior to and after the collapse, is constant within each subperiod. Similarly, the demand for money is constant during the subperiods 0 5 t 5 T and t > T. In order to further understand the effects of the new policy and the anticipated speculative attack, it is useful to formally derive the results described above. These results imply that during the transition period 0 I t I T consumption is constant at some level E, while, following the attack, consumption is constant at some level 2, where, from the first-order condition for consumption (7), 2/z = [q(‘*)/q(r*
+ p)p
< 1.
(15)
Similarly, using the same notation to differentiate between the transition period and the post-collapse period, it can be seen from (4) that 732/t% = [k(r* + k)/k(T*)](Z/E) < e/E < 1, indicating that, at the time of the collapse, real balances fall proportionately more than does consumption. Using (15) in the resource constraint (ll), which under the new policy becomes T
I
{ 1 + v[ k(r*)]}c,e-rl’dt
0
+
-{l + v[k(r* IT
it can be demonstrated
ct =
+ p)]}c,e-F’dt
= (y/r*)
that
~5= tj-“‘{(y + r*fo)/[(l e = Q-1/y {(y + r*f,)/[(l
- e-“r)8 - e-“‘)6
+ LeT6]}, + e+T6]} ,
+ ff ,
tt
5 T > T’
(16)
where 9 = q(r*), 9 = q(r* + p), 6 = (1 + v[k(r*)]}g-“‘, and 6 = (1 + v[k(r* + ~)]}4-“‘. For the discussion that follows, it is useful to think of the demand for consumption at time as a function of the effective price of consumption 9t and the marginal utility of wealth A. These “X constant” demand curves are given by [A9(*)],-“’ as can be seen from (7). As discussed by MaCurdy (1981), the y constant demand curve is the natural extension of the permanent income theory of consumption to a situation where the price of consumption varies over time (as it does here because of the impending speculative
t
697
Kent P. Kinabrough attack). In such settings the marginal utility of wealth, like permanent income, as it is usually defined, summarizes all information about the future path of the economy that is relevant for current consumption choices. Only when the effective price of consumption is constant is the standard concept of permanent income adequate. Applying the A constant demand concept, it can be seen from (15) and (16) that the q-‘lr terms in (16) capture the substitution effects of the new policy. It is also apparent that the term in braces in (16), which equals A-“‘, captures the influence of shifts in the marginal utility of wealth on the height of the equilibrium consumption profile. Just like the standard notion of permanent income, this term determines the height of the consumption profile, and from here on (y + r*fo)/[(l shall be referred
Announcement
- e+‘)4
to as permanent
+ e+%]
income.
Effects
One issue that has not been dealt with so far, but that is an important feature of an anticipated speculative attack, is the shift in macroeconomic variables at the time the new policy that precipitates the collapse is announced. At the time the new policy is announced, the effective price of consumption remains at its initial level, 0 = q(e). Th e announcement effect of the new policy on the equilibrium consumption profile is thus a pure wealth effect capturing the shift in permanent income that is associated with the announcement. Consumption rises at the time the unsustainable policy is announced if permanent income rises, and falls if permanent income falls. Additionally, since the opportunity cost of holding money is unchanged at the time of the policy announcement, the behavior of real balances is proportional to the change in consumption. Note from (12) that consumption under the original, sustainable policy can be written as Q-l’v times permanent income under the original policy, which is given by (y + r*fo/[(l
- e-“‘)I,
+ e+‘h]
.
08)
As can be seen from (17) and (18), permanent income rises or falls with the policy announcement as B S 8. From the preceding discussion it follows that
696
Speculative
Attacks
and
as
esii,
(19)
where
6(r* + a,) s (1 + v[k(r* + IT,)]}~-I”. Two factors thus work to influence permanent income at the time of the policy announcement. On the one hand, the opportunity cost of holding money increases after the collapse, reducing the money-consumption ratio (the inverse of the velocity of money) and thereby raising per unit transactions costs during the post-collapse period (that is, v[k(r* + TV)] > v[k(r*)]). This works to lower permanent income at the time of the policy announcement and tends to reduce consumption and real balances. The magnitude of this effect depends crucially on the interest elasticity of demand for money l/q. On the other hand, after the collapse the effective price of consumption is higher (4 > 0). This discourages consumption after the collapse and serves to reduce the resources devoted to transacting after the collapse. As a result, permanent income tends to increase at the time of the policy announcement raising consumption and real balances. The strength of this effect depends on the intertemporal elasticity of substitution in consumption l/-y.4 The preceding discussion suggests that whether or not permanent income rises or falls at the time of the policy announcement depends on the relative magnitudes of the inter-temporal elasticity of substitution in consumption and the interest elasticity of the demand for money. To see the exact nature of the dependence of the shift in permanent income at the time of the announcement on these two elasticities, recall (19) and note that 6 Z 6 as 8’ 2 0. From the definition of 0(m) it can be seen that 8’ = q-l’Y[v’k’ - (1 + v)(k/ y9)]. Using the fact that k’ = -(l/v”) along with the functional form
Q is important to note that both factors influencing permanent income at the time of the policy announcement depend on transactions costs. As a result, the possibility that permanent income may rise or fall with the policy announcement does not depend in any fkdamental sense on whether or not transactions costs are “large. ”
699
Kent P. Kirnbrough for u(e) and Equations (4), (6), and (12), it can be shown that 0’ = l~q-“~[(l/q) - (l/y)(l + &‘I, where p = r*k~/(y + r*fo is the permanent share of seigniorage in national income under the sustainable policy. This, in conjunction with (19) implies that
as WY)(l
+ PY’ 3 l/q
.
(20)
As will be discussed later, the share of seigniorage in national income is typically quite small so that, roughly speaking, (20) indicates that the impact of the new policy at the time it is announced depends on whether or not the intertemporal elasticity of substitution in consumption is greater or less than the interest elasticity of the demand for money. If the intertemporal elasticity of substitution exceeds the interest elasticity of the demand for money, permanent income rises when the unsustainable policy is announced and consumption and real balances also increase. If the reverse is true then the announcement effect of the unsustainable policy is to reduce permanent income and lower consumption and real balantes . In contrast to the announcement effects of the unsustainable policy, and as can be verified from (12) and (16), the post-collapse level of consumption is always less than the level of consumption under the sustainable policy regardless of the relative magnitudes of the intertemporal elasticity of substitution in consumption and the interest elasticity of the demand for money. This can be seen by noting that 6 can be written as y + r*fO times the inverse of the expression in square brackets in (21). Since the term in square brackets is a weighted average of two terms greater than 1 + 6, 2 < E. Intuitively, the inter-temporal substitution effects of the policy unambiguously outweigh the wealth effects. Since the opportunity cost of holding money also rises at the time of the collapse, the post-collapse level of real balances must also be lower than the level of real balances under the sustainable policy. The upper panels of Figures 1 and 2 illustrate the two possible time profiles for con700
Speculative
Attacks
bg0
T
t
bg0
-Q-
WYU
Figure 1. + PY > I/?
sumption and real balances following tainable policy.
the enactment
of the unsus-
The Balance of Payments Accounts Another important issue relating to the effects of unsustainable policies and anticipated speculative attacks concerns the behavior of the various balance of payments accounts. To begin, note from (20) that, as a first approximation, if the intertemporal elasticity of substitution in consumption is greater than the interest elasticity of the demand for money, real balances rise at the time of the policy announcement while if the reverse is true real balances fall. If discrete jumps in domestic credit do not occur, the Central Banks stock of international reserves may increase or decrease when the 701
Kent P. Kimbrough m.
lil ii
m t
(l/Y)(l
Figure 2. + PY <
‘i I I I I I I
.
;
I I
b
T
t
WI
unsustainable policy is announced. In fact, if the intertemporal elasticity of substitution in consumption is less than the interest elasticity of the demand for money, the unsustainable policy will be characterized by two attacks on the Central Banks stock of international reserves: the first attack occurs at the time the unsustainable policy is announced, and the second attack results in the collapse of the fixed exchange rate. In practice, these jumps in international reserves are likely to be spread out over a short period of time following the announcement of the new policy. However, after this brief adjustment period, the balance of payments will exhibit a chronic deficit up until the time of the speculative attack and the collapse of the exchange rate regime. These results are 702
Speculative
Attacks
summarized by the time profiles for international reserves shown in Figures 1 and 2. It can also be demonstrated that the behavior of the current account depends on the intertemporal elasticity of substitution in consumption and the interest elasticity of the demand for money. The current account is defined by ca, = y + r*ft - (1 + v[k(r* + ~FJ]}. From this and the fact that the consumption profile under the sustainable policy is flat, it follows that prior to the policy announcement the current account is balanced in all periods. At the instant the new policy is announced the current account is y + r*fO - (1 + v[k(r*)]}E. It follows from (20) that the current account deteriorates or improves at the time of the policy announceif the intertemporal elasment as (l/y)(l + p)-’ 2 l/q. Intuitively, ticity of substitution in consumption is large enough, permanent income rises with the policy announcement and the current account deteriorates. The decline in the economy’s stock of foreign bonds causes the current account to continue to deteriorate throughout the entire transition period 0 5 t 5 T. When the collapse occurs, the current account immediately becomes balanced and remains balanced thereafter. Alternatively, with a sufficiently low intertemporal elasticity of substitution in consumption, the policy announcement lowers permanent income and improves the current account. In this case the current account steadily improves until it once again becomes balanced at the time of the collapse. Figures 1 and 2 illustrate the two alternative paths for the current account. One implication of these results is that the current account is not a reliable indicator of an unsustainable mix of domestic credit and exchange rate policies. Under certain conditions the current account may actually improve when an unsustainable policy is enacted and continue to improve until the collapse takes place. Looking at things through the current account window may thus fail to signal the impending speculative attack and collapse. However, the chronic loss of international reserves through the balance of payments will accurately reflect the unsustainable nature of the policy. Speculative Attacks and Utility It is evident that, in the case illustrated by Figure 2 where the intertemporal elasticity of substitution is low, and hence (l/y)(l + p)-’ > I/q, utility falls upon anticipation of the speculative attack because consumption declines in all periods relative to what it would have been under the sustainable policy. However, when the intertemporal elasticity of substitution is high, it is not readily apparent 703
Kent P. Kimbrough that utility is reduced by the unsustainable policy since consumption is actually higher up until the time of the collapse. To formally derive the effects on utility of the increased rate of domestic credit growth underlying the anticipated speculative attack, note from (1) that in the presence of an anticipated speculative attack U = (1 - e+r)[C’-Y/r*(l
- r)] + e+T[2’-Y/P(l
- y)] .
Taking the derivative of this expression with respect to the rate of domestic credit growth and evaluating it at the initial equilibrium under the sustainable policy (where E = 2 so that the XT/+ term drops out of the resulting expression) yields dU/+
= (P/r*)[(l
- e+r)(fZ/a~)
+ e+‘(S/f!+.)]
.
The change in utility is the marginal utility of consumption, P, times the present value of the permanent change in consumption or the change in the “average” height of the consumption profile. However, from the budget constraint, T
(1 + 6)Ee-“‘dt
+
-(I + B)2ee-“‘dt = (y/r*) IT
it follows that in the neighborhood 6 = 8 and E = 2).
LO - e+‘)(dE/+J where 6 = u[k(r*)] falls by
NJ/+
+ e+T(dE/dpJ]
+ fo ,
of the initial equilibrium
= -u’k’EeweT/(l
+ i$ < 0 ,
and 8 3 u[k( r* + IL)]. This implies
= -u’k’E1-Ye+T/r*
(where
that utility
(1 + 6) .
This drop in utility, which occurs in all cases, is proportional to the present value of the rise in transactions costs associated with the unsustainable policy triggered by the rise in the rate of domestic credit expansion. The Timing of the Speculutiue Attack All that remains to fully characterize the impact of an anticipated speculative attack is to determine the actual timing of the 704
Speculative
Attacks
attack itself. When the attack occurs the Central Banks remaining stock of international reserves will be depleted and, unable to defend the fixed exchange rate S, the government will allow the domestic currency to float. Money market equilibrium at the instant of the collapse thus requires that k(r* + h)Z = l&ewT/S,. Since consumers correctly anticipate the collapse, the exchange rate cannot jump at the time of the collapse; if it did, unexploited profit opportunities would exist. Therefore, ST = S must also hold at the time of the collapse. This, (16), and the condition of money market equilibrium imply that the timing of the speculative attack is determined by the condition
k(r* + PXY+ r*fo)l(* - 6) = e@[(l - e-“r)(l
+ S)(o/f!j)“’
+ e+“‘(l
+ S)] ,
(21)
where 6 = v[k(r*)], i7 3 v[k(r* + CL)], and use has been made of the fact that (10) implies that at Do = S(ti - ba. Equation (21) implicitly determines the time of the collapse as a function of the model’s parameters and state variables at the time of the policy announcement. In particular, the time of the collapse depends on the rate of domestic credit creation under the unsustainable policy and on the initial stock of international reserves. For purposes of illustration, consider the effects of changes in the rate of domestic credit creation, CL, on the timing of the speculative attack. As can be seen from (21), given T, an increase in i.~ lowers money demand at the time of the collapse while simultaneously raising the supply of money. The direct effect of an increase in the rate of domestic credit expansion is thus to create an excess supply of money at the time of the collapse. In order to maintain equilibrium, the time at which the collapse occurs must adjust. Changes in T influence both money supply and money demand at the time of the collapse. It is readily apparent that an earlier collapse reduces the supply of money at the time of the collapse thus tending to restore equilibrium in the face of an increased rate of domestic credit creation. If (l/y)(l + p)-’ > l/q, then I + 8 - (1 + fi)((i/~) I” > 0 and an earlier collapse raises permanent income and the demand for money at the time of the collapse (the term in square brackets in [21] falls). This reinforces the equilibrating tendency on the supply-side of the market and hence, when (l/y)( 1 + p)-’ > l/q, the collapse occurs earlier in 705
Kent P. Kimbrough response to an increase in the rate of domestic credit creation (that is, I~T/+L < 0). However, if (l/y)(l + p)-’ < l/q, an earlier collapse lowers permanent income tending to create an even greater excess supply of money at the time of the collapse. If this effect is strong enough to offset the supply-side effect of an earlier collapse, it is possible that the collapse may occur at a later date when the rate of domestic credit creation is increased (that is, aT/ap > 0). Note, however, that the equilibrium time profiles for various macroeconomic aggregates may be as shown in Figure 2, yet an increase in domestic credit growth may nonetheless hasten the collapse. The conditions for a rise in the rate of domestic credit creation to postpone the collapse are more stringent than simply (l/y) (1 + p)-’ < l/q. Similar results can be shown to characterize the impact of the initial level of international reserves on the timing of the collapse. Typically a larger stock of international reserves will postpone the collapse (dT/dbg > 0). However, if the intertemporal elasticity of substitution in consumption is low enough, it is possible that larger international reserve holdings may hasten the collapse (dT/dbg
< 0).
4. Concluding
Remarks
The economic effects of an anticipated speculative attack and exchange rate regime collapse brought on by an unsustainable mix of domestic credit and exchange rate policies have been studied here. The approach has been to use a maximizing model with the demand for money motivated by a transactions technology which implies that increased money holdings reduce the transactions costs associated with consumption goods purchases. It has been demonstrated that the effects of an impending speculative attack depend crucially on two margins through which the forward-looking behavior of rational agents manifests itself: the intertemporal elasticity of substitution in consumption and the interest elasticity of the demand for money. Formally, it has been shown that the impact of a speculative attack depends on whether (mw
+ PY’ 2 l/q
>
where l/y is the intertemporal elasticity of substitution in consumption, I/q is the interest elasticity of the demand for money, 706
Speculative
Attacks
and p is the share of seigniorage in national income. However, evidence presented by Fischer (1982) indicates that seigniorage as a share of national income is typically quite small; usually in the range of I-3% with many countries having smaller shares and very few having shares greater than 4% of national income. Therefore, roughly speaking, the results of the paper indicate that the effects of an anticipated speculative attack depend on whether or not the intertemporal elasticity of substitution in consumption is greater or less than the interest elasticity of the demand for money. If the intertemporal elasticity of substitution in consumption exceeds the interest elasticity of the demand for money, an anticipated speculative attack raises consumption and real balances on impact, and results in an immediate and continued deterioration of the current account until the time of the attack. In contrast, if the intertemporal elasticity of substitution in consumption is less than the interest elasticity of the demand for money, an anticipated speculative attack lowers consumption and real balances on impact, and results in an immediate and continued improvement of the current account until the time of the attack that brings about the collapse of the exchange rate. In addition, in this case there may also be an attack on the Central Banks stock of international reserves at the time the unsustainable policy is announced. It is useful to contrast these results to those of other models of speculative attacks grounded in maximizing behavior. Calvo (1986, 1987) motivates money via a continuous time cash-in-advance constraint. This approach sets the interest elasticity of demand for money to zero. Calvo’s results can thus be viewed as a special case of those obtained here. In essence, his use of the cash-in-advance constraint to motivate money amounts to assuming that the inter-temporal elasticity of substitution in consumption exceeds the interest elasticity of the demand for money. The policy triggering the collapse that Calvo studies is different than the one studied here. He considers a situation where at t = 0 the rate of crawl of the domestic currency (that is, the rate of inflation) is temporarily reduced while domestic credit growth is maintained. Like the policy studied here, it is excessive domestic credit growth that gives rise to the collapse. If Calvo’s policy is examined using the framework of this paper, it is straightforward to show that his results hold only when (l/y) (1 + p)-’ > l/q. Another maximizing approach to studying speculative attacks has been to put money directly into the utility function. Obstfeld (1986) does this under the assumption that the utility function is 707
Kent P. Kimbrough separable in consumption and real balances. As a consequence, unlike the setup used here, Obstfeld’s model does not predict any real effects from an anticipated speculative attack. Claessens (1988) drops the separability assumption and extends Obstfeld’s framework to allow for real effects associated with speculative attacks. However, he does this by assuming that the instantaneous utility function is CobbDouglas, which amounts to constraining the interest elasticity of demand for money to equal one. This perhaps overstates the interest sensitivity of the demand for money. More to the point, as a result of constraining the interest elasticity of the demand for money to equal one, Claessens fails to notice that what is crucial for determining the impact of an anticipated speculative attack is the magnitude of the intertemporal elasticity of substitution in consumption relative to the interest sensitivity of the demand for money. The model presented here can thus be viewed as nesting other maximizing models that have been used to study speculative attacks, In view of this, it is important to know how useful the increased generality of the setup employed here might be in practice. One way to gain perspective on this is to ask what the empirical literature has to say about the model’s two crucial parameters, the intertemporal elasticity of substitution in consumption and the interest elasticity of demand for money. Hansen and Singleton’s (1983) estimates suggest an intertemporal elasticity of substitution in consumption of no less than l/2 and perhaps much greater. However, Hall (1988) has recently presented evidence that leads him to conclude that the intertemporal elasticity of substitution in consumption is “unlikely to be much above 0.1 and may well be zero.” Both of these studies use U.S. data. Regarding the interest elasticity of demand for money, Boorman’s (1972) survey paper suggests that this elasticity ranges from 0.1-0.2 to 0.35-0.75 depending on whether one uses short-term or long-term interest rates. Friedman and Schwartz (1982) estimate the semielasticity of demand for money to be around 10. For nominal interest rates between 5% and lo%, this implies an interest elasticity of demand for money in the range 1/2-l. These studies also use U.S. data. Blejer (1978) and Khan (1977) look at South American countries and report semielasticity estimates that are consistent with an interest elasticity of demand for money in the range of 0.1-1.5 (for nominal interest rates in the range of 20-308, which are reasonable for the period and countries they study). These studies indicate considerable overlap in the range of estimates for the intertemporal elasticity of substitution in consumption on the one hand and the interest elasticity of the demand 708
SpecuZative Attacks for money on the other. In conjunction with the model presented here, this suggests that, in practice, anticipated speculative attacks may not be characterized by a similar response of macroeconomic aggregates such as consumption, real balances, and the current account in all countries at all times. Received: December 1990 Final
version:
January
1992
References Blanco, Herminio, and Speculative
and Peter M. Garber. “Recurrent Devaluation Attacks on the Mexican Peso.” Journal of Political Economy 94 (February 1986): 148-66. Blejer, Mario I. “Black-Market Exchange-Rate Expectations and the Domestic Demand for Money.” JournuZ of Monetary Economics 4 (November 1978): 767-73. Boorman, John T. “The Evidence on the Demand for Money: Theoretical Formulations and Empirical Results.” In Money Supply, Money Demand, and Macroeconomic Models. Edited by J. T. Boorman and T. M. Havrilesky, 248-91. Northbrook, IL: AHM Publishing, 1972. Calvo, Guillermo A. “Temporary Stabilization: Predetermined Exchange Rates.” Journal of PoZiticaZ Economy 94 (December 1986): 1319-29. -. “Balance of Payment Crises in a Cash-in-Advance Economy.” journal of Money, Credit, and Banking 19 (February 1987): 19-32. Claessens, Stijn. “Balance-of-Payments Crises in a Perfect Foresight Optimizing Model.” Journal of Znternational Money and Finance 7 (December 1988): 363-72. Feenstra, Robert C. “Functional Equivalence Between Liquidity Costs and the Utility of Money.” journal of Monetary Economics 17 (March 1986): 271-91. Fischer, Stanley. “Seigniorage and the Case for a National Money.” Journal of Political Economy 96 (April 1982): 295-313. Flood, Robert P., and Peter M. Garber. “Collapsing Exchange-Rate Regimes: Some Linear Examples.” Journal of International Economics 17 (August 1984): 1-13. Friedman, Milton, and Anna J. Schwartz. Monetary Trends in the United States and the United Kingdom. Chicago: University of Chicago Press, 1982. 709
Kent P. Kimbrough
Hall, Robert E. “Intertemporal Substitution in Consumption.” Journal of Political Economy 96 (April 1988): 339-57. Hansen, Lars Peter, and Kenneth J. Singleton. “Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns. ” Journal of Political Economy 91 (April 1983): 249-65. Khan, Mohsin S. “Variable Expectations and the Demand for Money in High Inflation Countries.” The Manchester School of Economic and Social Studies 45 (September 1977): 270-93. Kimbrough, Kent P. “The Optimum Quantity of Money Rule in the Theory of Public Finance.” Journal of Monetary Economics 18 (November 1986): 277-84. MaCurdy, Thomas E. “An Empirical Model of Labor Supply in a Life-Cycle Setting.” Journal of Politica Economy 89 (December 1981): 1059-85. Obstfeld, Maurice. “Speculative Attack and the External Constraint in a Maximizing Model of the Balance of Payments.” Canadian Journal of Economics 19 (February 1986): l-22.
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