Rational Destablizing Speculation and Exchange Invervention Policy

International Finance Discussion Papers

Number 157

December, 1979

RATIONAL DESTABILIZING SPECULATION AND EXCHANGE INTERVENTION POLICY

by

Matthew B. Canzoneri

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. Reference in publications to International Finance Discussion Papers (other than an

acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

Rational Destabilizing Speculation and Exchange Intervention Policy

by

Matthew B. Canzoneri*

Introduction

*

The U.S. interpretation of the current international monetary regime has been explained by Anthony Solomon (1978), Undersecretary of the Treasury for Monetary Affairs, as follows:

The basic philosophy of the new monetary system... is that international monetary stability cannot be imposed from without, but must be developed by countries from within, through the application of sound underlying economic and financial policies.

In line with that concept, our program for assuring a strong and healthy dollar relies on fundamental economic performance, not on market operations to hold or attain a particular exchange rate or maintain a particular exchange rate zone. We do recognize, of course, that markets can become disorderly, subject to great uncertainty, dominated by psychological factors and speculation. We have made clear that we are fully prepared to intervene in the markets to counter such disorders.

Controlling the fluctuations caused by non-fundamental or psychological factors has become a major goal of U.S. intervention policy. All sorts of things are thought to be capable of generating these fluctuations. News of political or institutional change is often blamed, as is the reporting of new figures on inflation rates,

trade balances, oil imports, etc. Of course, these pronouncements

I would like to thank J. Gray, D. Henderson, M. Dooley, P. Minford,

K. Rogoff and D. Waldo for useful discussions of the material presented here; however;.the views expressed here are solely those of the author

and do not necessarily represent the views of the Federal Reserve Board or other memebers of the staff.

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may well herald the working of fundamental economic forces that would, unless offset by Solomon's sound underlying policies, eventually affect exchange rates; however, these fundamental economic forces

can not be the cause of the "knee jerk" market reaction that often

follows such announcements. Instead, these fluctuations are usually attributed to psychological or speculative factors. The recent instability in gold markets has also been identified as a possible source of instability in exchange markets, even in quarters that had previously admitted little or no casuality in that direction _/

Most of the existing models of destabilizing speculation postulate speculators that make some sort of systematic prediction error.=! By contrast, exchange market participants are generally thought to be rather efficient in their use of information. The purpose of this paper is to present a characterization of destabilizing speculation that does not depend upon systematic prediction errors, and to examine its implications for a stabilizing intervention policy.

The notion that rational speculators might build "psychological" factors into stock market prices, and be right in doing so, is not new. Keynes (1936) makes the colorful analogy to a beauty contest in which the judges are allowed to bet upon the outcome. For each speculator-judge, guessing what the other judges will take as a sign of beauty becomes as important as discerning the innate pulcritude of the individual contestants, In fact, it becomes all important unless the winning judge is awarded some claim on the actual beauty of the winning contestant. In the same way, exchange market speculators

must consider both the fundamental soundness of individual currencies

and the capital gains and losses that will be caused by the actions

of other speculators. If one speculator expects others to react to changes in, say, the price of gold, then he too will react to changes

in the price of gold. And if enough speculators behave in this manner, exchange rate movements will indeed reflect movements in the price of gold, fulfilling the speculators’ expectations and needlessly disrupting exchange markets.

This kind of phenomenon can be captured in rational expectations models. Shiller (1978), for example, has noted that extraneous variables can become a part of the solution to such models, causing _ more uniqueness problems of the type discussed by Taylor (1977), and indirectly by Sargent (1973) before him. However, the rational expectations literature has tended to view these uniqueness problems as a nuisance; they are universally assumed away. Here the extraneous -variable problem will be focused upon as a characterization of ‘rational, but destabilizing, speculative behavior.

. It turns out that there is a simple intervention policy which, if imposed, will minimize the disorder created by this kind of speculation; unfortunately, there may be some reason to believe that it is not the kind of policy that is usually envisioned. In particular, it will be shown in a model. incorporating the "new" monetary approach to exchange rate determination that a policy of

leaning against the wind only makes matters worse. The correct

policy, according to this view, is one of accomodation: a depreciation should be met with an increase in the money supply. This unlikely policy conclusion is not a general implication Of the characterization of destabilizing speculation that is being proposed here; rather, it is an implication of the (admittedly strong) version of the "new" monetary approach that has been adopted for illustrative purposes,

A counterexample will be provided (in an appendix) in which the correct policy is to lean against the wind. On the other hand, the present characterization of destabilizing speculation is somewhat unconventional in that certain uniqueness problems can not be simply assumed away in a manner that has become standard in the rational expectations literature, and this fact does seem to affect the way that models incorporating the "new" monetary approach work. So a

more conventional stabilization problem will be presented for comparison; it will be seen that when the conventional assumptions are imposed to achieve uniqueness, a policy that leans against the wind can stabilize the exchange rate in models incorporating the "new"

monetary approach to exchange rate determination.

1. A Model of Rational Destabilizing Speculation

A simple small country framework can be used to illustrate this characterization of destabilizing speculation. This may not be considered the appropriate setting for a discussion of U.S. inter-

vention policy, but it does yield a clear and unemcumbered view of

the destabilizing speculation and its implications for monetary policy. “The basic conclusions will be true in more complicated settings.

Consider a small open economy whose output is a perfect sub-

stitute for foreign goods. Purchasing power parity implies

(1) Py = PH +e,

where p and p* are the domestic and foreign prices of the home product (in logs), and e is the home currency price of foreign exchange (again in logs); the foreign price is fixed at p* which, for simplicity, will. be set equal to zero, Domestic producers can sell all they want at the going world price; the (log of) output is fixed

at its full employment level, y. (2) y.2y=s_ +8, (1/07) t ° 1 e

Full employment output is inversely related to the asymptotic variance of the exchange rate, 025 exchange rate volatility is assumed to impede trade flows.2! The existence of on in (2) plays no role in the present

characterization of destabilizing speculation. It turns out that the monetary authority can limit the size of 0°, and the "fixed versus flexible rate" literature suggests that there is some reason for its doing so. The inclusion of oe in (2) is the simplest way of modeling

this benefit in the present setting.

pre cee aa a ARN ote fRabrang ene. -

So where does the destabilizing speculation come from? In this model it originates in financial markets. There are only two assets, home money and foreign exchange. Home residents must divide their wealth between these two assets, and they are assumed to hold the entire home money stock, They hold home money for transactions purposes and for speculative gain. The real return on foreign money is zero (since p* is fixed), while the real return on home money is a

-e ). From the portfolio manager's point

random variable, “(Qa t

of view, the mean of this return is

(3) E [Ce - e] = -( e,)

“re[ 2 7 Mt

and its variance is

2 2 * - — + —_- = - = (4) EC eet Ce pe 7 eT = Eeleeay -— Ceeriel” = tere . a : *2 E ° ° ral ] is the conditional expectation operator; eet | rand eee (t are the "rational" mean and variance of e based upon information

tt1L 4/

available at the end of period t.— It will turn out that e -e

ttl t has a normal conditional distribution; so a risk averse portfolio

manager's demand for home money will be of the form

i eS co a vere + ny, where a, y and n are positive constants.

Notice that conditional means and variances appear in the money demand function (5) while asymptotic variances appear in the supply curve (2). It is often asserted that financial markets respond more quickly to new developments than do goods markets, or that labor markets are encumbered

by long-term contracts. Differential conditioning of moments is one

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way of modeling this asymmetry. Each period, portfolio managers watch prices as they form and chose the portfolio that maximizes their expected utility

of return. The output supply structure, on the other hand, does not respond

to day by day occurances; it responds only to changes in asymptotic moments.

Equations (1), (2) and (5) determine e and Yee Using (1) and (2)

t? Pe to eliminate P. and ye in (5), the money market equilibrium condition can

be solved for an equation that determines e.:

a

(6) af odo is C= Yer)" ©

It will turn out that en \t is not time dependent, so c may be regarded as a constant.

If the money supply is fixed at n, then (7) e. =e e=zc+m=mn- ty

is an obvious solution of (6). Here there are no random elements in the model, so (6) becomes an ordinary (but unstable) difference equation with a stationary solution at e. Others have discussed Stability and uniqueness problems that arise in the deterministic case;2/ the present paper focuses upon the "extraneous variable" problem.

The extraneous variables in this case might be a series of proclamations, forecasts, or statistical reportings by governments or by private agents, or they might be changes in prices of irrelevant goods like (in the present

model) gold or oil. In short, they can be anything that one speculator

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might think other speculators will take as a leading indicator of exchange rate movements. These variables will be modeled as sequence {u,} of serially uncorrelated, normally distributed random variables with zero mean and variance 0, (Adding serial correlation merely complicates the mathematical expressions without changing the conclusions in any important way; the case of serially correlated extraneous variables will be discussed in an appendix. It may also be interesting to note that the entraneous variable need not be stochastic; that is, it may be perfectly predictable. Things do work out somewhat differently in this case; it too is treated in an appendix.)

Can this extraneous variable become a true, self-fulling leading indicator? If so,

(83) e =ets w

.u t i=0 ite-i

should also be a solution of equation (6) (for some appropriately chosen values of the coefficients e, Wo» Wy> Wo recede Equation (8) says that the extraneous variable actually becomes a part of the solution at some time

t = 0. If equation (8) is indeed a valid solution, then

ttl - t

(9) eer pe WEF ES MMe MOF Mite . isl i=0 and t (10) a fe 7 Pe * Seopa” MMe

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Substituting (10) into (6) (with nm, set equal to nm), that

expression becomes

= t _ (11) eam tot OE so aay wus

and (8) is consistent with (11) if

or

wi) w. = [(a + 1)/a]*wy

So if the coefficients ©, Wo? Wyoe++ are ‘chosen to satisfy the constraints in (12), then (8) will be a valid solution of the model. Speculators’ use of the extraneous information embodied in u. can indeed cause the exchange rate to fluctuate needlessly. And this case of destabilizing speculation can not be attributed to systematic prediction errors since speculators’ expectation formation is "rational." The extraneous information can become a true leading indicator.

Before going on, there is one technical matter that should be

disposed of. Earlier it was asserted that the conditional. variance

BS

a

ene would turn out to be time independent, so that c could be treated as a constant. It is time to verify that assertion, and this is easily done. From equations (8) and (9).

. . a fone wu ” * and va e” = woe’

(13) e ~e. an ttl t#1|t Ot tht Ou

There is a uniqueness problem associated with this characterization

of destabilizing speculation. Note that (8) is a valid solution of

equation (6) for any set of coefficients that satisfy (12), but (12) places no

- ll =

restriction upon Woe There is a difference equation generating the w

i? no initial condition is given. Put another way, with restrictions (12),

but

equation (8) becomes

_7 t i (14) e,=e + Wor 3} Ug i=0

e=m+tc=m + yin. - 1 B=@tl1)/a

and (14) is a valid solution of the model for any choice of Woe There are several ways of comparing the solutions corresponding to

different choices of Wo: One approach is to compute the unconditional

variance of e.:

2 22.t 2i (15) of =won 6 e u™”, t i=0

For any W, # 0, this variance increases over time, and it increases without

bound since 8 > 1. As long as speculators continue to view u, as a leading

indicator, the bubble will continue to grow.2/ (And there is no obvious

‘reason for them to quit taking u, as a leading indicator, for expectation. formation is rational and u, has actually become a part of the solution.) Another approach is to compute expected future exchange rates. For-

warding equation (14) to period T and taking the conditional expectation

| (6) e

. t oi ~ T-t_

tle SF Mor Mra pe 8 FM x =ptu. + po tu + +u

t 7 0 » °°" t

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erie is the exchange rate that portfolio managers espect to obtain T-t periods from now. Unless a highly improbably canceling has occured (so

that x, = 0), e

t ‘rt 7 = as T+ if w) #0. That is, for all choices of

Wo other than zero, the exchange rate is expected to explode or implode. This uniqueness problem always arises in rational expectations

models that include expectations of future prices in the equations

that determine the current price.&/ In the past, it has been handled

in various ways. Sargent's (1973) "no speculative bubbles" condition

ruled out any solution for which the price is expected to explode or

implode, while Taylor (1977) chose the solution that minimizes the

asymptotic variance. Here, either criterion implies a choice of Wo

equal to zero, but setting Wo equal to zero in (14) results in equation

(7). The Sargent-Taylor solution excludes the extraneous variable

altogether; they simply assume that speculators collectively choose

to ignore the leading indicator .2/ In the present context this

amounts to throwing the baby out with the bath water. In what

follows it is assumed that Wo is some number other than zero, but the

indeterminacy remains,

. It should also. be noted that e depends upon w In addition, this

0°: ' long-run on average exchange rate depends upon the degree‘of risk aversion (y).and the volatility of the leading indicator (o,).. II: ' Policy Tiiplicatitons ©

Does there exist a practical intervention policy that will stabilize

exchange rates against this ‘kind of speculative. disturbance? The answer

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depends upon what the monetary authority is assumed to know. It also depends upon what is meant by the word stabilize. There would seem to be two stabilization goals worth considering. The monetary authority could try to minimize the asymptotic variance of the exchange rate,

the variance that curbs production and trade flows. (See equation

2

tt1|t that

(2).) Or it could try to limit the conditional variance é

lowers portfolio managers’ (ex ante) welfare. . If the policy maker thinks he has identified the leading

indicator that is causing the disturbances, then he may be

tempted to try to stabilize the indicator itself instead of inter-

vening in exchange markets, (If o can be set equal to zero, then

both oe and e-

te1|t will vanish.) Recent flirtation with the idea of

trying once again to stabilize the price of gold would appear'to stem from this kind of thinking 2/ The present author has serious doubts about this approach. First, an unsuccessful stabilization effort

would only draw attention to the indicator and probably make matters worse. Second, there are many extraneous variables that could concievably generate speculative disturbances; there may be several operative at the same time. Discerning the popular fancy may be difficult, and once again, mistakes only draw attention to potential’ sources of instability. Finally, stabilizing all of the potential sources of instability would, even if possible, be prohibitively expensive. The present paper considers instead policies that intervene in exchange

markets directly.

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In particular, consider simple policy rules of the form (17) m =o + g(e, - e) : 11/°12/ . where g is an as yet undetermined policy parameter.—~ -~— Substituting (17) and (10) into (6) results in

=7 _ of . (18) e,rmted4 py [ows + (g a)w, Ju,

i=0 i

and (18) is consistent with (8) if e and the Ww, are chosen to satisfy

(19) e=mt+e e=m+ yo - ty or i w, = aw, + (g - a)w, w, = [8 - (g/a)] Wo B= (a+ 1l)/o

With these restrictions, solution (8) becomes

(20) e, = € + wort oie - (g/a)l*u,_,

e m+ yor = ty

where Wo is once again undetermined, and the variances in question become

“2 _ 22 (21) etl |t = Woe, _ 2 2.t 2i Te 7 “Ow ino ~ (8/0) wo" of _ Ou

e I-@- g/a)2 Gf | 8 = (g/a)| <1)

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The choice of the policy parameter g has no effect upon the

conditional variance e 3 portfolio managers can not be helped in

tt1|t

13/ tea : 2 an ex ante sense.— However, the unconditional variance ow, can

t radically modified. If g is chosen to make | B - (g/a) | < 1, then o. no longer grows without bound, and the asymptotic variance is t

finite. Speculative disturbances will continue to grow, but not without bound. The optimal intervention policy and the solutions it

implies are:

(22) g = g* where

o

i} o + = (=)

t Ot 2 2 *2 22 k= = = = e = w g of atl oe, Fe, er41| t WOeu

This policy rule reduces all of the variances to the conditional variance. It may be interesting to compare these solutions to the Sargent-

Taylor solution discussed in the last section. If the optimal policy

is imposed, all of the solutions satisfy Sargent's "no speculative

bubbles condition": that is, for all Wo

e+w.u =e forallT>t

“rit ~ OT|t

and the exchange rate is not expected to blow up. Technically, all

of the solutions in (22) qualify as Sargent solutions; as Taylor pointed out, Sargent's condition does not guarantee uniqueness. Taylor's procedure of chosing the solution that minimizes a still requires

setting w, equal to zero; his condition always achieves uniqueness,

0 and it rules out destabilizing speculation.

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It is important to note that the stabilization policy described above is not the conventional lean against the wind policy that is usually envisioned. In fact, it is clear from (21) that leaning against the wind (or setting g <0) actually increases variances. Policy must be accommodating (g > 1) to stabilize exchange rates; if the exchange rate depreciates, the money supply should be increased and more then proportionately.

This policy prescription may sound strange at first, but it is not difficult to see why it works in the present model: If the exchange rate rises above e, it must be forced back down; an appreciation must be engineered. "Rational" portfolio managers will foresee this appreciation and raise their demand for money; the (real) supply of money must be increased to accommodate this new demand. Put another way, the monetary authority must accommodate the demand for money that is consistent with the expected appreciation or depreciation that moves the exchange rate in the desired direction.

Finally, it should be noted that portfolio manager's risk aversion makes the average exchange rate e depend upon Wo» no matter what policy is instituted =4/ It will fluctuate in a manner that the policy maker can not predict as bubbles come and go. This fact

may or may not in itself be a problem, but it will certainly make

the success or failure of any stabilization effort hard to document.

III. A More Conventional Stabilization Problem It should not be thought that the accommodating policy prescription

of the last section is the general policy implication of the new monetary

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approach to exchange rate determination when expectations are assumed to be rational. In fact, the results of the last section run counter to the implications of more conventional rational expectations models as well, More conventional models often imply that leaning against the wind can be a beneficial policy, An example is provided below.

How is the present model unconventional? The standard practice with rational expectations models is to rule out solutions for which the exchange rate is expected to blow up. The speculative bubbles described in the first section would be ruled out by assumption," and the stabilization problem discussed in the second would never come up. It is in this sense that the model outlined in the first section is unconventional.

It should be emphasized that the present characterization of destabilizing speculation does not depend upon this explosive nature of expectations. A model is provided in Appendix 1 in which solutions incorporating extraneous information satisfy Sargent's "no speculative bubbles" condition, S

The model outlined above does exhibit explosive behavior, and it has affected the qualitative nature of the appropriate stabilization policy. To see why this is so, it is helpful to consider a more con-

ventional stabilization problen.

Suppose the money demand function is respecified as follows:

(23) m-e = ~oCe ile - e.) -ct u,

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Here the random variable u, is a part of money demand. It represents a fundamental economic force, and not an extraneous variable, In fact, extraneous variables will play no role in this example; in keeping with tradition, the explosive solutions will be ignored. The

money supply rule is

(24) m= h + ge,

and the purpose of this example is to show that negative values of g will lower a. Leaning against the wind stabilizes the exchange rate. Substituting (24) into (23) and rearranging

(25) e, = Poe .s lt + $6(h +c) - Gu

(l+@-)7

Ss Mi

Forwarding (25) j periods and taking the (conditional) expectation, (26) ae | t = Woe ie lt + $(h + c)

since u, is not serially correlated. If (26) is successively substituted

into (25) (with j set equal to 1, then 2 and so on), the result ist8/

2D e, = Lim Wo) euyn) , + 8h + c)e" (a)* - gu

00 1=0 t

Note that @ <1 for all g <0, so the only solution for which Curle

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does not blow up is

O(h + c)(1 - fa)7t = gu

(28) e. = t o” = 6707 = o7/(1 t+a- 8)? e u u

Negative values of g clearly stabilize the exchange rate.

Why does eliminating the explosive solutions change the qualitative nature of the stabilization rule? The present author is not entirely sure, but some intuition may be gained by considering two non-stochastic varsions

of the model. In particular, the money demand function can be rewritten as

! - = =» - - (23) m7 & ate e.) c

or

(23)" mo -e, = - a(e - e,) -c

Equation (23)' is the perfect foresight version of the model, Rearranged, (23)' is a difference equation generating next ‘period's exchange

rate;

Cy = [at 1)/ale, - (1/4) (c + m,)

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As is well known, this difference equation is unstable; however, the policy rule

m, =ht ge,

changes the dynamics to Cry 7 (1+ Cl = e)/ale, - (1/a)(c + h)

and it takes values of g greater than unity to stabilize the equation. This is reminicent of the results in section II; in fact, setting g *= ge =q +1

reduces the equation to

ey 7 7 C/a)(e + h)

which is comparable to (22).

In (23)", the expected future exchange rate is fixed at e, a constant. A decrease in the money supply produces an appreciation of the home currency; this is the only way to create the expectation of a depreciation that is consistent with a lower demand for money. In this model, leaning against the wind works. Shocks that would tend to depreciate the currency can be offset with a tight money policy, This is reminicent of the more conventional stabilization problem just considered.

With (23)', the expected future exchange rate does all of the adjusting to equate supply and demand for money in the current period. With (23)",

the forward rate is fixed, and the spot rate must do the adjusting. Clearly,

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the rational expectations models (with either (17) or (23)) fall somewhere

in between; both e, and e adjust to equilibrate the market at time

t ttl |t t. However, it appears that the conventional rational expectations model, with its long-run expectations pinned down, behaves more like the deter-

ministic model with (23)".

IV. Conclusion

The extraneous variable problem associated with rational expectations models has been suggested as a characterization of destabilizing speculation that does not depend upon systematic or perverse prediction errors. Speculators build extraneous variables into their forecasts of future exchange rates simply because they expect other speculators to do the same; their collective behavior then ratifies the forecasts and exchange rates fluctuate for psychological or non-fundamental reasons.

There is a simple intervention policy that-will stabilize the exchange rate, but it is an accommodating policy that seems to run counter to conventional wisdom. In response to a depreciation, the monetary authority should increase the real supply of money, thereby creating a. level of demand that is consistent with an appreciation of the currency. This policy prescription appears to stem from the fact that without an effective stabilization policy the exchange rate would be expected to blow up. The present characterization of destabilizing speculation does not depend upon this explosive nature of expectations; it could have been explicated in a more stable setting. However, the most straightforward modeling of the new monetary approach to exchange rate determination does exhibit this explosive behavior

unless speculative bubbles are simply ruled out by assumption,

References

Aoki, M. and M. Canzoneri, "Reduced Forms of Rational Expectations Models," Q.J.E., Feb. 1979, pp. 59-71.

Black, F. "Uniqueness of the Price Level in Monetary Growth Models with Rational Expectations," J.E.T. 7, 1974, pp. 53-65.

Brock, W. "A Simple Perfect Foresight Monetary Model," J.M.E., April 1975, p. 133-150.

Coes, D. "Firm Output and Changes in Uncertainty," A.E.R., March 1977, pp. 249-251.

Friedman, M. "Inflation and Unemployment," J,P.E., June 1977, pp. 451-472,

Keynes, J. The General Theory, Macmillan, 1936,

Kouri, P. "The Exchange Rate and the Balance of Payments in the Short Run and in the Long Run," Scand. J. of Econ., May 1976, pp. 280-304,

Minford, P. Substitution Effects, Speculation and Exchange Rate Stability, North Holland, 1978,

Sargent, T. "Rational Expectations, the Real Rate of Interest and the Natural Rate of Unemployment," B.P.E.A., 1973.

Sargent, T. and N. Wallace, "The Stability of Models of Money and Growth with Perfect Foresight," Econometrica, No. 1973,

Sargent, T. and N. Wallace, "Rational Expectations, the Optimal Monetary Instrument and the Optimal Money Supply Rule, J.P.E, 83, April 1975, pp. 241-255

Sandmo, A. "On the Theory of the Competitive Firm Under Price Uncertainty," A.E.R., March 1971, pp. 65-73.

Shiller, R. "Rational Expectations and the Dynamic Structure of Macroeconomic Models," J.M.E. 4, 1978, pp. 1-44,

Solomon, A. "Exchange Market Developments and U.S. Policy," presented at

the International Herald Tribune/Forex Research Ltd. Conference on Managing Foreign Exchange Risk, 1978.

Stern, R. The Balance of Payments, Aldine Publishing Co., Chicago, 1973,

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Taylor, J. "On Conditions for Unique Solutions in Stochastic Macroeconomic Models with Price Expectations," Econometrica 45, Nov. 1977, pp. 1377-1381.

Tobin, J. "Liquidity Preference as Behavior Towards Risk," Review of Econ, Stud., Feb. 1958, pp. 65-86.

Footnotes

1/ The New York Times (October 3, 1979) quotes Anthony Solomon as saying the gold fever "not only reflects concerns about inflation, but becomes

an engine of inflationary expectations." The Times goes on to point out that this was the first time an American official had formally acknowledged a spillover from gold market activity to exchange markets.

2/ See, for example, Stern's (1973) discussion of stabilizing and destabilizing speculation.

3/ Criticisms of flexible rate regimes are often based upon this assertion. Stern (1973) discusses some of the issues involved in this long literature; the present model is not sufficiently rich to explore them. One way of justi-

fying the inclusion of o in the present model is to allude to the work of

Seandmo (1971) and Coes (1977). They show that risk averse firms will contract in response to greater price uncertainty. Friedman (1977) has also argued that the natural rate of employment depends upon the variance of prices.

4/ The rational expectations hypothesis asserts that speculators' subjective views about the moments of future random variables are identical to the model's own mathematical expectations conditioned upon information available

when the predictions are made, and e are conditioned upon

“etl | t tt1 | t

all variables dated t or earlier. Shiller's (1978) survey of the rational expectations literature is excellent background reading for the present discussion,

5/ a and y need not be positive if income effects are large; see, for example, Tobin (1958). The postulated signs are generally considered to be the relevant ones.

6/ See Sargent and Wallace (1973) or Kouri (1976). There are similar problems in models that add random disturbances to the structural equations; see Sargent (1973).

7/ Why is it that risk averse portfolio managers continue to hold home money while the variance of the exchange rate is increasing without bound? This apparent discrepancy is resolved when it it recalled that portfolio managers re-evaluate their portfolios each period and base their decisions upon the conditional variance of the exchange rate which is statimary; see equation (13).

8/ See Shiller (1978) or Aoki and Canzoneri (1979).

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' 9/ There. haye been a number of attempts, generally in a deterministic “framework, to rule out "unstable" solutions on microeconomic grounds (see Brock (1975) or Minferd (1978), but it is hard to see how their arguments pertain to finite lived utility maximizers like the portfolio managers postulated in the present model, Yt should be noted that the explosive naturé of the present model is not a necessary feature of this characterization of destabilizing speculation. Appendix 1 provides a model in which solutions incorporating extraneous information satisfy Sargent's "no speculative bubbles" condition. It will also be seen that when the optimal stabilization rule is imposed in the present model,

solutions reflecting extraneous information can still occur, but they will satisfy. Sargent's condition,

10/ Various rumors of such a policy were reported by the press about the time "of the IMF meeting in Belgrade (October, 1979). George Willis of the U.S. Treasury told the American Mining Congress (Los Angeles, Sept. 26, 1979)

that official sales of U.S. gold stocks "responded to conditions in the

gold markets last year, which had contributed to the adverse psychological atmosphere in the foreign exchange market."

11/ Here again, the information available to the policy maker has been limited

to make the problem interesting. The optimal full information policy rule is obviously ,

me Tm OC ye 7 ee)

If it were implemented, equation (6) would reduce to (7), and once again both variances would vanish. However, to calculate the expected rate of depreciation, the policy maker must have identified the extraneous variable,

and he must know the value of Woe See equation (10). This would not appear

to be the revelant case. On the other hand, private agents are assumed to be able to calculate Cut |e? and it is not entirely clear why the policy maker

should be at an information disadvantage.

12/ Some would prefer a feedback rule of the form m. m+ gle.) - &)

where the policy maker is not assumed to know the current value of the exchange rate. However, the dating structure on expectations in the money demand function implies that portfolio managers are able to watch the exchange rate as it settles upon its equilibrium value and modify their demands accordingly. Here, it is difficult to see why the policy maker should have an information disadvantage; nevertheless, this case is treated in an appendix. The mathematics becomes more complicated, but it turns out that the results are basically unchanged, provided that q > 1.

- F3 -

13/ This is essentially the familiar Sargent-Wallace (1975) result. Policy rules can not affect price prediction errors unless they incorporate some information advantage. In the case of unconditional variances, they do precisely that; the feedback term used data that was not used in the (unconditional) prediction.

"14/ At first glance, this fact would seem to make the policy rule (17) “untenable; by assumption, the monetary authority does not know Woe

However, (17) can be expressed in a form that is usable.

B ut

m+ gle, - e)

h + Be, hem- g

The monetary authority sets g and h; setting g equal to g* will again result in solution (22). The Wo that obtains will determine e

and the average money supply m fht ge).

15/ It will be recalled that lim en}, was not finite unless Wo was set equal Toe | to zero.

16/ This solution technique originated with Sargent (1973).

Appéndix'1:- An Example of Non-Explosive Destabilizing Speculation

There have been a number of attempts to justify Sargent's "no speculative bubbles" assumption on microeconomic grounds; see Brock (1975) or Minford (1978). It is difficult to see how their arguments pertain to finite lived utility maximizers like the portfolio managers postulated above. However, the jury is not yet in on this issue, so it is worth noting that the explosive behavior described above is not an essential feature of the present characterization of destabilizing speculation. A model is presented below in which solutions incorporating extraneous information satisfy Sargent's "no speculative bubble" condition; that is, the exchange rate is not expected to explode or implode. The non-uniqueness will of course remain.

This counterexample is essentially Taylor's (1977). Suppose real money balances are a factor of production, so the supply curve (2) is replaced by

(2)" y, =y +8,(m, ~ p,)

With (2)" in place of (2), (6) becomes

(6)" e =c+ale

t - e,) +m

tt1|t t

c = c/(1-ns,) a =Ea/ (1-n8,)

and a solution to (6)" can be found by replacing c and a with c and a

in (14):

t =

’ : (14) e, ret Woes a & Deng

g -= (om 41) fo el- (ns-1)/a

The exchange rate expected to obtain T-t periods hence is

_ “T=t_ ~ “tel ' = " — (16) ere e+tw, 8 x, x = 8 Up +8

+ wee tu wy

and the unconditional variance of e. is

(15)' oe =w t

Now suppose 1 < nS 5 < 1+ 2 a, so that [B | <1. In this case,

_ _ 22, 2 =e and oo = Woo, / 4 )

lim erie e

To for all values of Woe All of the solutions satisfy Sargent's "no speculative bubbles" condition, and the unconditional variance approaches a finite bound, the asymptotic variance oe

So here is an example of non-explosive destabilizing speculation. Once again, extraneous information can become a part of the solution, but the explosive, snow~balling effect has been eliminated. In the main body of the paper, it was shown that a properly designed monetary policy can play the same role that real balances in the production functions plays here; either renders the model "background stable" and creates uniqueness problems even when Sargent's condition is imposed, Taylor (1977) demonstrated this result for real

balances in the production function; F. Black (1974) demonstrated it

it for accomodatiye policy rules, —

Tt may be interesting to note that leaning against the wind is an appropriate policy in the present model if 1 < NS 5 K1lt+a. Demonstration of this fact is left as an exercise for the interested

reader,

‘Appendix ‘2: Serially Correlated Extraneous Variables

Serial correlation does not seriously alter any of the reguits in the main text. It does cause the optimally controlled solutions (the

solutions corresponding to (22)) to cycle,

To see this, consider three equations

Ct) e, =e + aCe le e,) + m, (II) UL, = Puy + ey (IIT) nm =m + gfe, - e)

(I) ts equation (6) in the main text. (II) explains the correlation

structure of the extraneous variable; e, is a serially uncorrelated

normal random variable with zero mean and variance a? (III) is the monetary policy rule.

Once again, a solution of the form

_ t =e+t+ (IV) e,=e D290 W, Uy

will satisfy (I). If (IV) isa valid solution then

t+1

met ’ OF Fino Ma MEtea|t

Seth |t

ttl

*"ertwo-. + . et wopu, + Ply Meu a

e+ (wp + Ww, )u, + ie] Waa tpeg

and

ne = Gwe + wy - Wu, + Tier

- Ww, Serle 7 Se ju

Substituting this expression and (IIT) into’ (1) and gathering like terms, a . . | . | (v) e,zetmt Tow, - (ow, - (a-co-g)w. Ju, + Un LOW, - (o-g)w,Ju._s

so the restrictions on e@ and the Wy become

(VI) erct+n w, = [8-p-(g/a) Iw, Wi “ [8-(g/a)]*w, for all i > 2 & = (atl)/a

Once again, there is no restriction on Woe And as before, if g=0, the speculative disturbances will continue to grow as time passes. The same feedback rule can be used to limit this instability;

setting .@ equal to g*(=08), the solution becomes

=e -~ u (VII) e, tet wi (p/a)wiu |

and e. has a finite asymptotic variance (assuming lp <1). The basic difference between (VII) and (22) is that serial correlation in the leading indicator induces persistence or cyclical effects in the controlled

solution,

‘Appendix’3;. Lagged Feedback Rules

Some may object to the instantaneous feedback in the policies discussed in the main text. However, it turns out that policy rules of the form A) m =mt ge, -e)

MQ) a

yield very similar results provided that a is greater than one.

see this, consider once again the equations

1. This restriction on a is perhaps believable. a is the partial derivative of the log of money demand with respect to changes in the The interest elasticity of demand 9 is equal

"interest rate," e -e

ttl t° to a « @ where é@ is the "average" rate of change of the exchange rate; so a > 1 is equivalent to > é, The interest elasticity of money demand

must be greater than the average rate of change in the exchange rate.

2. It may be interesting to note that F. Black (1974) gets the same

restriction.

(BL oe, ac t orale ne) tm,

(c) e, et Lieg Wy Meg

(8B) is the reduced form for the exchange rate (equation (6) in the main text), and once again it has a solution of the form (C).— The now.familfar procedure produces the following restrictions on

e and the Ww! -

(DP) e=ctan

Wao 7 Bwy4, + bw, = 0 h = g/o

Again, there ts no restriction on Woe The solution to the difference equation takes the form

od 4 (E) w, = cA; + CoA,

where cy and Cy are constants’ that depend upon the initial conditions .

Ww, and w, (= Bw), and 4 = Lig+@7-an)*] and i, = 518-6 7-4n)*)

The asymptotic variance of the exchange rate is

2_ 2,9 2_ 2 4 4 2 pey2t i (F) of a Tet 7% [ey 27a," EA + 2eyeo%, oy J

so the policy problem is to find an h that will make the difference

equation stable and the A 4 real, Then the asymptotic variance will be finite.

Suppose first that h <6 27143 then the Ay are real. Let x =

@? - 4h) / 3 then if a > 1 (so that 58 <1), My and ro are functions of

x as shown below;

Any choice of x in [o, x) will produce a stable difference equation for the

Wy Since W and thus cy and c, are not known, it is not possible to tell which point in [o, x) is preferable. Now x is in {o, x) if g =«h is in (1,

2 (1+a)“/4a]; so once again the optimal g is greater than one.

Appendix 4: Non-stochastic Extraneous. Variables’

If the extraneous variable is exactly predictable then future values of the extraneous variable will also Be included in the solution,

With this adjustment the results are much the same as before,

To see this consider the eauations

(T) e ec +a(e - e,) +o,

tt1|t

u

(iI) m, a constant

t

(III) we tu, Uses}, a non~stochastic sequence. and consider solutions of the form

—_ t © CTV) ee et Ey wt tye eee where the Wy and the y, are undetermined coefficients. Here future values of the extraneous variable have been added to the solution.

Using the now familiar procedure (and noting that ret = eu

since all of the Gee are known to speculators at time t), the interested reader will be able to show that (IV) is a valid solution if the coefficients

are chosen to satisfy the restrictions.

(V) wy = [G40)/a]* w

on it

g 7 [e/ (te) }*y,

w = Vv. fo) (0)

Qnce again, ® can he any number... — Yt should be noted that the future values of the extraneous vartable must be included in the solution. ‘The interested reader will

also Be able to show that tf (VI) sotrt ow, t te tao “i “tei

ts proposed instead of (VI) as a solution form, then the restrictions on the coefficients become Ww, = 0, The extraneous variable can not be

built into the solution unless future values are considered.