19

(i) A larger depreciation of the exchange rate under uncertainty than
would occur under perfect foresight (assuming the same change in asset
supplies)

(ii) A depreciation in the forward rate, even though expectations are

unchanged.

Note that even if the deficit is fully financed by bonds, there
is still an exchange rate depreciation, though a smaller one than if it
is money financed. These results demonstrate that-unlike the perfect
foresight case-money and bond supplies both affect spot rates. Further,
they show that for given expectations, it is changes in the stock of debt
(and not its composition) that affect the forward rate. This is
plausible since the exchange risk is present for money and bonds; if
agents are to be induced to hold, for example, more country i currency,
they will wish to reduce the bonds (denominated in that currency) that
they hold. Hence, the (real) rate on those bonds must rise, compared to
other assets. Changes in the spot rate are dominated by money supply (or
demand) movements, as differential interest rate movements enter
indirectly; but the changes in the forward rate (or interest rates) are
dominated by the total stock of debt (given expectations).

Further, note that there is no necessary correlation between
the spot and forward rates (or forward premium). Tight monetary policy
and a temporary) deficit can lead to an appreciation of the spot rate,
but a “orward rate depreciation and increases in the real interest rate
on domestic bonds (versus other bonds). A deficit and less restrictive

monetary policy (Me > 0) implies spot and forward depreciation, and


Appendix
Derivation of Comparative Static Results
(51) of the text can be rewritten as:
(1A) a(y, &) = EL(v2(Kyy)4/Y - vAy(v2y + viyan2y = 0
For notational convenience, define:
(2A) A = Lv@(Ky)/¥ ~ vq; B = (vey + vi)
Let J, = (ad/9z), z = Y, K. Then:
(3A) [Yedy] = ~ EL((V8/y)(K/¥)1/ 1B + (1 = adYV2-a)Bt - 21 < 0

since
(4A) EL(Yv2/B) (AeB* - 1)] > 0

since (AB*~ 1) 2.0 as (v2/vl)2 (y/K)/¥ and (Yv2/B) increases as

wen) )increases.

2 Vya-1
(5A) CKdx] = (1/y)ELV (K/Y) BJ > 0.
Hence:
(6A) (a¥/8K)(K/Y) = - [UyK/(JyY)] > 0.
Let: .

(7A) ¥ = agks (JyY)¥ + (JyK)K = 0, implies:
(8A) ELV ¢(K/¥)1/1((1-ay)/y)B = ay (1 - a) YAIB*~ 2] = 0

For a, = 0, the LHS of (8A) is positive; for a, = 1, it is negative
(a < hy, Thus, ay e(0,1) fora< 1.

Rewrite (8A):
(9a) EL(2) ¢¢K/¥)/¥(ay(1 ~ a) - (1 - ay)/y))V?
- V(ay(1 - @)¥ + ((1 - ay)/y)(K/Y) 7) ya" > 27 = 0
Since 0 < a, < 1, the coefficient of v2 must be positive:
(10A) 1 > a, > [1 + y(1 - «)1 -1 which is the result of (52). This seems
to be the closest vey hs that can be placed on a).

If have identical distributions:

(11A) EL (v2 - vty) .(y2 + ylyoly = 0 by symmetry.


13

The problem is most severe when interest rates are non-
stationary as would arise, for example, if one country grew more rapidly
than the other, or the relative supply of bonds to money changed
systematically over time. If we wish to find a closed-form solution, we
must specify the model so that the probability distribution of interest
rates is stationary. This entails assuming that the stock of nominal
debt (currency plus bonds) denominated in each currency follows a first
order Markov process (or, less plausibly, a stationary distribution),
whereas all other disturbances are stationary. Define:

i

i i). yi iyi = epiyyi
Ie + BE) = Me(L + o4)s94 = (B/E)

(35) D

= (M

(36) Kye CLy(1 + o2)/( + ob) T = pnPa2(q2yMca + By /mta Cat)" + of)

In (35), } is the total stock of currency i denominated
nominal debt, and a is the proportion of bonds to currency ((1 + aj)
‘is the fraction of debt that is monetized). Ke embodies other sources of
uncertainty (plus a), specifically those which affect currency demands.
Assune:

(37) Dea, = Uy Des 7 = 12

*
(38) Ke =K q,


23

static results can be derived. Totally differentiating (51) and

simplifying yields (see Appendix for details):

(52) Yea, Kil>a>fM+y1-aT > 0, ac]
1 1

(a = 1 implies a, = 1). By definition:

(53) K = (2 = a4) + nq? - a2) + E(a92/(1 + 92)) - (atv + 94)

(54) BL = + (ag' (1 + g!))

Com + gtBE/(1 + gt)

From (32), (42) and (52):

(55) 6, = V+ a - a = ayl( - 0) + lq? - gy]

Na = ay atic + 9}))
- (RECT + ayo?) + 21 = a B21 + 92)

gl 1
+ ((M,.01 + aig )+g

1
to

(56) fF, = LC + WY - ky/yd + DE - 02

(57) Fp e, = 1% - RY

where (F, - e,) is the change in the forward premium.
Since g’ is assumed constant over time, prices in each
country grow at the (random) rate of debt expansion, and nominal interest

rates are constant. The difference in real returns on bonds is given hy:


(vi) all bonds are one period bonds that pay a market
determined nominal interest rate r'(t).

(vii) country 1's currency is taken as numeraire; e(t), the
market determined exchange rate, is the number of units of
country 1's currency (dollars) needed to purchase one
unit of 2's currency (marks).

These four assets are the only stores of value. If there is no
specific transactions demand for money, country i money and bonds will be
perfect substitutes (r" (t) = 0). Even if there is a transactions demand
for currency, the exchange rate may be indeterminate (Kareken and
Wallace, 1981) if the currencies are considered perfect substitutes. ! To
avoid this problem, we postulate per capita money demands for each

country as follows: 2,3

(1) om = atptlagly a 4 rly; nyo: i = 1,2

In (1), time subscripts are suppressed for simplicity; a' isa

money demand parameter, pi is the domestic currency price of the single
i .
good, q is per capita output, and nm! per capita money demand. Assuming

commodity arbitrage:

2 1

(2) Pe = e,Pe

All agents in a country are identical, and only the young hold

money. Hence, aggregate money demand is:

i isi,i
(3) ME = Naps

i iv-y.;
i ("a tts t= 12


F2

10/ As usual, a multiplicity of solutions can arjse. From (20), one
possible solution is: Cray = fey (+ rt y/ + rey) for all (t + 1)
realizations. Solving bickwards® this yields: :

Ing = IEE (mm, CC + yyy,

We = ome (Md )J, which is explosive. We rule out this solution, and
séek one which only depends upon current variables and expectations.

11/_ Since there are no transfers, the assumption on debt growth implies
assuming nominal government puchases grow at a random (constant mean)
rate, and expected real purchases are constant over time. Ry allowing
taxes on the young, the (seemingly remote) possibility of budgetary
surpluses could be handled as well.

12/. For a = 0, the utility function in (8) is not defined. However, the
utility function: U = Ince + o1nSt 44 yields behavioral rules identical to
those derived in the text when a 1s set equal to zero. Even for a= 0,
the disturbances (other than debt expansion) must be stationary. If, for
example, output levels follow a first order Markov process, we cannet
derive a closed form solution.

13/. Within the context of the analysis, any change in (g', q', x') must
be viewed as a surprise. However, even if the change at t had been
anticipated at (t - 1), this would not affect period t equilibrium,
though the anticipation would affect, of course, the equilibrium at

(t - 1). The affect of anticipated permanent changes can easily be
derived by first showing how the actual change affects the equilibrium
distributions, and then using (for t - 1) the transitory analysis of
section III by viewing the (t - 1) variable value as a transitory change
from its permanent level.

14/ Clearly, it is not appropriate to ask how changes in the variability
of 4] affect spot rates since an incregse in the variability of e
(given its mean) will, increase EL (e441) 1, and will also change the?
variability of (e441, ).

15/ Another reason for considering the reciprocal (vi = (uty-4y is if it
is real spending that is stochastic PhiGhay = (hy - 1) ob; or

Chay = (1 - VE Eal(q?y)na + gl)(1 + r4)"Y1, Hence, a mean preserving
spread of real government spending corresponds to a MPS of Vbats not

1
Yew (given rly,

16/ Essentially, this says that a positive risk premium is likely to be
‘associated with an appreciating currency (i.e., a currency that is
expected to appreciate). This result is consistent with “stylized
facts," such as shown in Figure 1 of Wyplosz's article (1983, p.124).


20

higher (relative) real domestic interest rates. Since exchange
expectations are given, relatively higher domestic real interest rates
(than abroad) are compatible either with a currency that is expected to
appreciate or depreciate. In sum, temporary budget deficits should lead
to higher domestic interest rates (relative to abroad), but the movement
in the spot rate will largely be dominated by monetary policy.

Finally, consider the effect of sterilized intervention that is
assumed transitory. Clearly, the efficacy of the intervention depends on
how future expectations are altered, as well as how current asset
supplies change. In the spirit of this section, assume the intervention
does not alter expectations concerning future spot rates (current
suppiies of assets to the private sector change, of course). To be
consistent with the structural model, this means that expectations
concerning future asset supplies are also unchanged.

If future asset supplies are unaltered, and if nominal
government spending is also unchanged, then the profits (losses) of the
intervention must be refunded to domestic residents. Thus, suppose the
central bank of country 1 purchases Ca units of country 2 bonds with
B= e, BP) units of its own bonds. At (t + 1) the position is
liquidated; the net proceeds from the transaction (ey Be(ler?) -

e, Be (14rt)) are refunded (lump-sum) to domestic taxpayers (the "old").
Clearly, if domestic residents (in the aggregate) increase their holdings
of domestic bonds by BE (=e, 82) and reduce their holdings of foreign bonds
by x, they will remain in portfolio equilibrium (including the

transfer/tax due to intervention). In essence, because of the transitory


monetary (debt) expansion, the efect of uncertainty is to lead to a
depreciation of the currency for which real bond yields are larger.

The explanation is immediate--the higher interest rates (or
forward rate) leads to a decreased demand for currency, and hence a
depreciation of the spot rate. From (55) - (59) the effects of permanent
changes on spot and forward rates can be deduced, and compared to treir
perfect foresight results. Unlike the case of transitory disturbances,
permanent changes in money demand (via w or q'), or permanent changes in
the composition of the debt (given its level) will have different
quantitative effects under uncertainty than under perfect foresight.
This is because these changes affect future spot exchange rates and
hence, under risk aversion, have different affects on the forward rate
change which is needed to induce agents to hold the existing asset
stocks.

Comparing (61) to (62), it can be seen that the effect of
uncertainty is to reduce the sensitivity of the spot (and forward) rates
to shifts in money demand or supply (for given debt levels}. Under
perfect foresight, a 1% increase in demand for currency 1 (via increases
in wt or q?) will lead to a 1% spot (and forward, due to stationarity)
exchange rate appreciation, whereas under uncertainty the proportional
spot (and forward) appreciations will be less. Comparably, a change in
money supply (via a permanent open market operation) will have a smaller
proportionate effect under uncertainty. Pure debt expansion, given the
composition of the debt, will have the same proportionate effect under
uncertainty and perfect foresight.

Furthermore, note that under uncertainty there is no direct

correlation between movements in the spot. rate and forward premium (or


analysis that follows, we assume this to be the case. The qualitative
results for our analysis and for the case in which a= 0 and other
disturbances (such as money demand, output levels, etc.) are presert are

identical, so we omit the latter case to save space.

Specifically, we assume: (i) the nominal debt in each country
follows a first order Markov process, as in (37): and (ii) that agents
assume current values of other variables (g', a‘, ) will prevail
forever. Given these assumptions, we inquire how permanent changes in
these variables, or in the random rate of debt expansion, affect
equilibrium values? Under the above assumption, the portfolio rule
(39) is:

2 2 1

ye ety ve ew yy =o

(80) EL(v t+] t+ ttl t+1 t t+

1
t+1 *

Since the ve are stationary, the solution to (50) implies We» and hence

Yeo (since K is constant) are constant. Dropping the time subscripts,

(50) may be rewritten as an implicit function determining Y (in terms of
K, and the distributions of V'):

(51) atyyk) = ev? - pK OM yyy + viyerty = 0.

Further, since pageycy.k)1 > 9, and Jintacy. Kyl" <0, a soluticn Y"(k)

exists. Finally, since Jy(Y, K) < 0 at J = 9, the solution is unique.

While no analytic solution to (51) is possible, comparative


References

Barro, R., 1974, “Are Government Bonds Net Wealth"; Journal of Political
Economy, 82, 1095 - 117.

Dornbusch, R., 1976, “Expectations and Exchange Rate Nynamics", Journal
of Political Economy, 84, 1161 - 76.

Driskill, R. and S. McCafferty, 1980a, "Speculation, Rational
Expectations and the Stability of the Foreign Exchange Market",
Journal of International Economics, 10, 92 - 102.

> 1980b, “Exchange Rate Variability, Real and
Monetary Shocks, and the Degree of Capital Mobility under Rational
Expectations", Quarterly Journal of Economics, 95, 577 - 86.

> 1982, "Spot and Forward Rates in a Stochastic
Model of the Foreign Exchange Market", Journal of International
Economics, 12, 313 - 32.

Frankel, J., 1979a, "The Diversifiability of Exchange Risk", Journal of
International Economics, 9, 379 - 94.

>» 1979b, "On the Mark: A Theory of Floating
Exchange Rates Based on Real Interest Rate Differentials", American
Economic Review, 69, 610 - 22.

Grossman, G.M. and A. Razin, 1983, "International Capital Movements under
Uncertainty", NBER Working Paper Series #1075, Cambridge, MA.

Hansen, L.P. and R.J. Hodrick, 1980, "Forward Exchange Rates as Optimal
Predictors of Future Spot Rates", Journal of Political Economy, 88,
829 -53.

Helpman, E. and A. Razin, 1982a, "A Comparison of Exchange Rate Regimes
in the Presence of Imperfect Capital Markets", International
Economic Review, 23, 365 - 88.

> 1982b, "Dynamics of a Floating Exchange Rate
Regime, Journal of Political Economy, 90, 728 - 54.

Kareken, J. and N. Wallace, 1981,, "On the Indeterminacy of Equilibrium
Exchange Rates", Quarterly Journal of Economics, 96, 207 - 22.

Musse, M., 1982, "A Model of Exchange Rate Dynamics", Journal of
Political Economy, 90, 74 - 104.

Siebert, A., 1982, "The Risk Premium in the Market for Forward Foreign
Exchange", International FInance NDiscussionPapers #211, Roard of
Governors of the Federal Reserve System.

Wyplosz, C. A., 1983, "The Exchange and Interest Rate Term Structure
Under Risk Aversion and Rational Expectations," Journal of
International Economics, 14, pp. 123-39.


v7

canstent, a temporary 1% change in Ms will change a by (1/(1 + y))%, the
same result as under perfect foresight. Again, the explanation for this

rests on the constancy of e (and its distribution). As we shall see,

t+]
a permanent change in the composition of the debt (given its time path)

will rot have equivalent effects.
Proposition III: An open market operation that is perceived to be

transitory will have the same (percentage) effect on spot exchange rates
under uncertainty and perfect foresight.

Next, consider how a (surprise) budgetary deficit affects the
exchange rate. In order to concentrate on transitory changes, we assume
expectations concerning next period exchange rates (and prices) are
unaltered. Consistency requires that (the probability distribution of)
future asset supplies, and their composition, are unaltered. Hence, we
assume agents expect an offsetting surplus (lower deficit) next period,
as well as unchanged debt composition.

Under perfect foresight, the (surprise) deficit will have no
‘effect on spot and forward rates if it is bond financed, and will have
the same effect as an open market purchase of bonds if it is financed by
temporary money creation. For uncertainty the results differ. Since the
debt does not follow a random walk, (39) does not hold. Instead, rewrite

(20) as:

2
tot+l

2

(45) EL(1 + (Dee, /0E)) TPL M(E(Dee, 45 )/ (DLW, )T = 1) = 0

Since the distributions of (e Pay) are assumed unaltered,

t+1?
W,can-conceptually-be solved for in terms of (nt /02). While no analytic

solution is possible, comparative static results can be ascertained.


Using the assumption of constant relative risk aversion,

define

iL i2 2 i2 i2 il 1 il il
(19) “t = e,Tb, d+ ry) +m Thy / 0b, (+ ri) +m T te!
In (19), y! represents the ratio of next period's net wealth held in

t
marks to dollars, evaluated at the current exchange rate. Using (19),

(15) simplifies to:

(20) ECCT + (Yhepay Zep) tag) Ueean # re) - ee + rd)

"
°

(20") ELC + (Wey ys/eg))e (Pegg) ACCT + ety) = (1 + o¢4))I = 0

Since (20) (or(20')) is independent of agent specific
variables, portfolio decisions (Y,) are the same for all agents. Using
the budget constraint and the first order conditions, the agent's
‘behavioral rules may be simplified to:

ai

(21) ch = Cas + HD]

-1 -1

2
+Y, 0 + 7 ]

(22) 4 1)

-1
Cox) “LC + rb)


32

variable. Since the above factors can lead to appreciation or
depreciation in the spot rate, there is no clear connection between
movements in the spot rate and in the real interest differential between
countries (or their bonds).

The current situation of large U.S. budgetary deficits, a
strong (and appreciating) dollar, and high real interest rates in the
U.S. versus its trading partners can be accounted for within the context
of the model. If the current budgetary deficits are viewed as
transitory, to be followed by lower deficits (than abroad), anc if the
deficit is largely financed through bond issuance, then the overall
affect could be to lead to an appreciation of the spot rate. These
factors--increases in the proportion of debt that is accounted for by
bonds, and decreases in expected future deficits--will also lead to an
increase in the expected real return on domestic (versus foreign)
assets.

At the same time, much of importance has been omitted from the

“analysis. Since nominal assets are the only stores of value, one cannot
ask how exchange rate variability--as caused by expectations concerning
future asset demands or supplies--affects real investment, or compare the
levels of real investment under fixed and flexible exchange rates.
Similarly, since there are no wage contracts (denominated in nominal-or
any-units), one cannot ask how exchange-rate variability affects
employment, or compare the performance of fixed and flexible exchange
rates with respect to stabilizing employment. However, a model of this

type may be generalized to deal with these important issues.


29

premium or real interest diferential (which is zero). However, for a <
1, an “ncrease in © increases the forward premium and the relative real

return on country 1 bonds.

Proposition VII A decrease in the rate of country 1 debt expansion
(given its relative variability) will, if agents are risk averse: (i)
lead to a smaller percentage appreciation of country 1's currency than
would occur under perfect foresight; and (ii) lead to an increase in the
risk premium, or relative real interest rate, associated with country 1
assets 26/

Under perfect foresight, a 1% reduction in the rate of monetary
(debt) expanion will lead to a (y)% appreciation of the spot rate. From
(55) and (65), the spot rate under uncertainty will appreciate by less
than y% (it may, conceivably depreciate if a< 0). Hence, part (i) of
the proposition follows, and reaffirms our earlier results that
uncertainty reduces the sensitivity of the exchange rate to movements in
“exogenous parameters. Part (ii) of the proposition follows immediately
from (67) and (68). The increase in the relative real return on domestic
assets occurs because the variability of the real return on these assets
is increased by an increase in © (decrease in the rate of debt
expansion).

liice.,

On the other hand, a mean preserving spread (MPS) of V
an incraase in the variability of the real return on country 1 bonds, for
a given mean) will lead, as expected, to a spot depreciation of the
exchange rate and an increase in the relative real return (and risk

premium) on country 1 bonds. Let:


Next, consider the individual agent's optimization problem, We
assume:
(i) all agents have identical, homothetic preferences;
(ii) utility is an additive function of consumption in each period.
Assumption (i) is a standard trade theory assumption that
allows one to ignore the income distribution, and (ii) is a common
assumption in multiperiod models. Together, they imply:

i

(8) Uu(C. +1

Fishy) = (/ad0(CE)® + alStyy)81 a <1, a #0: > 0
where ch tsp yy) is consumption by the young in t (old in (t + 1)), 6 is
the rate of time preference, and a is simultaneously a measure of
relative risk aversion (1 - a) and intertemporal commodity
substitutability ((1 - a7). The assumption of constant relative risk
aversion greatly simplifies the analysis since portfolio composition is
independent of wealth.?

At t, the young choose current consumption and asset holdings
to maximize expected utility. Current prices, and the "true"
distribution of Play are known, but the realized values of Pay are not

known. Define:

(9) mas the per capita demand by the young of country i for currency j;

(10) be? as the per capita demand by the young of country i for country

j's bond.


The exchange rate is given by:

(4) eg = REAM LLO + ebay + 2)

(5) tye C2 (a2 ytd (qhyTM

In (4), i, is per capita money supply.

Under perfect foresight, interest arbitrage implies:
- 1 2
(6) 1 et Ca + ra + rd

where ead is also the one period forward rate at t. Using (4)

T
and (6), and assuming lig C(e(t + nj 190, B =[y/(1 + y)], the

exchange rate is given by:
(7) Ine, = f+ y)I, Fo {Cnt i, Ae Rt

t - “4 20 “ttictti tti

Thus, as usual, the current spot rate depends upon current, and
time t expectations of future, variables. Under perfect capital
mobility, real interest rates are equalized internationally, and bond
stocks do not affect the exchange rate. To focus on the role of

uncertainty, we use the ad hoc money demand functions (1) as a constraint

on optimizing behavior.


18

From differentiating (45), it can be shown that:

y 1 .
(46) We = 7a, (D4); 1> ay > aforarO: (if a= 1, aj = 1)
In (46), the notation % denotes the percentage change in X, (dX, /X,)

Substituting into (32) and using the definition of Y (H,) yields:

t

* we ag ne
(47) e, = [ME + Dey(1 - ag) (1 + y)
Using the expressions for the forward rate, and real interest
differential, their changes (around E(4,) = 0) are given by:
A “y

(48) f= (1 - ag) D,

(49) d€La,T = [(1 + rf )Pe/oeTEM,DE E(1/02,,)1(1 ~ ago}

For ay = 1 (risk neutrality), these are the same results as for
perfect foresight. In general, for risk aversion the effect on the spot
rate depends upon hoth money supply and bond supply changes, whereas the
movement in the forward rate depends only upon the level of the deficit.

From (47) and (48):

Proposition IV: Assuming agents are risk averse, a temporary increase in

debt (temporary, and offset, deficit) leads to:


> where f.

(15) EC(S4 ,)° Meg gy (1 # re) = eg (1 + re) V/PEyy)I =

ECCS) 71 + bea) ~ (1 + of ID = 0
(16) (1 + of.) = rk + ky k=, 2

In (14) - (16), ok is the realized real return on country k bonds, and
E(oS) is the expected real return. Under risk neutrality (or perfect
foresight) the (expected) real returns are equalized; for a < 1, there is
no such presumption.

Clearly, the introduction of a forward market will add ncthing,
provided there are no transaction costs. From covered interest

arbitrage:
1 2
(17) f= fe, (1 + ry /0 + ry]

t is the one period forward rate. Allowing forward positions

would imply:

(18) ELS 1) Cpa . Pad =0

which is equivalent to (15). As is well-known, the forward rate need not
be an unbiased predictor of the future spot rate. As noted by Siebert
(1982), the “risk premium" (E(ee 41) - f,) depends on the degree of risk

A + 1 +
aversion and the covariance between (Crap Peat): Even for risk
neutrality, a "risk premium" can arise, although the expected real

returns on bonds will be equalized.


24

(58) a, = EC + rEd(PE/Pty) ~ (1 + re y(Pe/Pe 1 = (1 +r? ytcvsKy Wh = vey

The change in E(4,). around E(A,) = 0, is:

1

1
tel )

M(Y = k)/y) #4 EVE

2
(59) dLe(ay)] = (1 +r )(v/K) YE
The above comparative static results allow us to infer how
changes in the exogenous variables affect equilibrium prices: however,
the level of these prices cannot be inferred without specifying the

distributions of wv, v2), For now, assume that yi and v? have
identical, independent distributions. Under this assumption, it is clear

from (51) that (see Appendix):

(60) Y2zlask 21; (Y/k) gl as K 21.
The results embodied in (55) - (60) allow us to make the following

inferences:

Proposition V: Under risk aversion, the expected real return on the two

bonds will, in general, differ-even if disturbances are symmetric.

From (60), it is apparent the expected real returns (given identically distributed
random debt growth) will be equal only if K = 1 -in essence, only if the two
countries are identical. In general, the real return on country 1's bond will be

higher if: (i) it is a larger country;


International Finance Discussion Papers
Number 232

January 1984

EXCHANGE RATE DETERMINATION AND
REAL INTEREST RATE DIFFERENTIALS UNDER UNCERTAINTY

by

Harvey E. Lapan

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
pub" ications to International Finance Discussion Papers (other than in
acknowledgment by a writer that he has had access to unpublished
material) should be cleared with the author or authors.


11

units, so that the tax liability of country i residents depends on the
stock of bonds denominated in that currency.® Total tax collections in

country i (T, are:

i
te)

(27) 4

— paipi
ter 7 (BE)

The government budget constraint is:

(28) M+ BL = My + BE

t-1 t-1 &

i si, oi
(Vt red a Tet PE

aan

i
= (Mey

i i,i
+ Bey) + (PES)
where G is real government purchases. Hence, net debt (currency plus
bonds’) issuance corresponds to nominal government spending. ? The per
capita tax liability (assumed to be a lump sum tax) is:
igo. ini yi, ny a wae yea
(29) Tea = 855 (TRYIN ); 54 j =0, i #4; 535 =l,i=j.
Combining the government budget constraint with the aggregate

consumer demands, and assuming all markets clear yields, after some

simplification:

yl e ytplyly lyn 1\-y
(30) = NPL AR(a, TC + ry)

2 _ y@pl,_2\n 2\-7,2
(31) eM = NPECqe) (+ re) dg


10

(23) x, = ELM + (egg /eg)) lip t/ha) (1+ ay

(24) G = Lap -((rgm.) + Teyy)/(PECL + red) ) -eg (Crem? + TEA V/(PLCL + red)
(25) (1 + rp )bt? = redal Cox, ead T+ Ty mf?

(26) (1 + r2)biZe. = Cy Pa cg yh) 70 +H T +e (THe - mi?)

Equations (22) - (24) are definitions, whereas (21), (25) and (26) are
the commodity and asset demands (together with currency demands, (11))
In essence, a is present discounted real income, net of tax liahilities
and money holding costs. Also, x reflects the expected real return (in

utility units) of current savings. Under risk neutrality, X, is one plus

t
the expected real return on bonds. Summing (25) and (26) across all
agents yields aggregate demand for each bond. Equilibrium in the bond
and currency markets yields current prices and interest rates, given
expectations.’

To complete the model, we must specify government behavior,
including how interest payments are financed. If there are no taxes and
(part of) the interest payments are monetized, there is a direct link
between bonds and exchange rates, even under perfect foresight. If all
interest payments are financed through further bond issuance, no steady-
state solution is possible, as nominal bonds (and the ratio of bonds to
money) grow over time. Thus, we asstime each government levies taxes (on

the "old") that just cover the interest due on maturing bonds. Further,

we assume each country issues bonds denominated only in its own currency


31

V) Conclusions

In this paper we have presented a general equilibrium model in
order to investigate how exchange rates and real interest differentials
are determined under uncertainty. Because the starting point of the
analysis was the assumption of money demand functions typically employed
in the Monetary Approach to Exchange Rate Determination, many of our
results are qualitatively similar to those predicted by the Monetary
Approach with perfect capital mobility.

Nevertheless, we have seen that the presence of uncertainty
does lead to some modifications of these conventional results. As seen
throughout the paper, the uncertainty and risk aversion reduces the
sensitivity of exchange rate movements to changes in exogenous
parameters, such as output levels, money demand disturbances, or
expectations concerning future monetary (and deht) expansion.
Furthermore, we have shown that it is the total stock of deht denominated
in a particular currency, and not just currency supplies, that affect
spot and forward exchange rates. For given money supply levels, an
increase in the stock of bonds denominated in a particular currency will
lead to a spot depreciation of that currency and to an increase in the
"risk premium" associated with holding that currency.

Furthermore, we have been able to investigate the determination
of the risk premium-or real interest differential-within the context of
the mocel. Generally, we have seen that the real return on domestic
bonds will be higher than on foreign bonds if (i) the domestic country is
larger, (ii) a larger fraction of domestic debt is in terms of bonds;
(iii) the expected rate of domestic monetary (debt) expansion is lower

than abroad, or if (iv) the domestic intertemporal price level is more


30

(69) vi =a+zts a, scalar, E(Z!) = 0, ef(z!)*> 0.

An increase in © corresponds to a MPS of wy, and hence of the real
return on country 1 bonds (given nominal interest rates). As shown in

the Appendix:
(70) (dY/de) > 0.
Consequently:

Proposition VIII: An increase in variability (a MPS) of the real return
on country 1 bonds leads, under risk aversion, to: (i) a depreciation of
1's exchange rate, and (ii) to an increase in the risk premium, and
relative real return, on country 1 assets.

However, care must be taken in interpreting this result; it
does not imply that an increase in the variability of the rate of debt
(monetary) expansion--that is, less predictability concerning future
nominal deficits--will lead to an exchange rate depreciation and a
higher "risk premium" for that country's currency. Due to Jensen's
inequality, increased variability of yh (debt expansion) raises the
expected real return on country 1 bonds (given nominal interest rates).
Thus, as shown in the Appendix, if agents are not "too" risk averse
(a > 0) a MPS of ul will lead to an appreciation of country 1's currency.
Consequently, in estimating how "variability" or "forecast errors" affect
exchange rates and real interest rates, great care must be taker in

specifying the source-and form-of the variability.


Ry assumption, the mo are not choice variables, but are

constraints imposed on the agent:

(an) mo = (oj oPLa (qh na + ely: 65 = 1. 4 = is bj 70 145
Each individual of country i receives an endowment (ap) when
young. In addition, he may pay taxes in the second period of life.° Let
TH be nominal taxes that a resident of country i pays (when old) in
country ;'s currency (ri = 0, i # j, if taxes are levied only in
domestic currency). It is assumed the nominal (not real) value of taxes

due is known ex ante.

The individual's budget constraints are:

i j il i1\,91 i2 , ,12),51
(12) a - Oy - C(m,. + be y/P 1 - Le, (m, +b Pd >0
il, 1 il _ sil i2 2 j2 _ si? 1 i
(13) PEAT + re) tomo - TET t epg fb + re) + me = Tea) - PeapStay 2 9

Assuming non-satiation, (12) and (13) hold as equalities.
. i 1 i . . 1
Given (4; Pio ep ri) and expectations concerning (Praye Cra)? the
agent maximizes expected utility (from (8)), subject to constraints (11)
-(13). Assuming an interior solution exists,” the first order conditions

imply:

(2a) (Ch) = get(st O11 + of Ts k= 1, 2


21

nature of the intervention, agents treat that part of the Central Bank's
Balance Sheet as "inside" wealth (or debt), and the sterilized
intervention is ineffective. It could be made effective by: (i) assuming
the proceeds/losses affect nominal government spending; (ii) transferring
the proceeds to the "young"; or (3) changing expectations, perhaps via a
"permanent" intervention. Otherwise, the sterilized intervention will

not have any effect.

Proposition V: Transitory sterilized intervention is ineffective if
Central Bank profits/ losses are rebated to taxpayers.

That completes our analysis of transitory disturbances. In the
next section we consider how “permanent" changes affect equilibrium

values.

IV) Permanent Changes and Exchange-Rate Effects

The analysis of the previous section was facillitated by the
assumption that future exchange rate and interest rate movements were
exogenous. A permanent change, however, will lead to shifts in the
distribution of these variables. Since an analytic solution for interest
rates (in terms of exogenous variables) is not obtainable, it is not
generally possible to derive comparative static results for permanent
changes.

An examination of (39) shows there are two cases for which
solutions can be found; (i) if Pear is constant, or (ii) if a= 02,

The nominal interest rates will be constant over time if the only source

of uncertainty concerns the rate of debt expansion. Hence, in the


3A

(22'a) mvt) = E re%2 qv2¥(1 + ((1 = a)/W)) + a3)
2

(am/dv!) < 0 for a > 0. Hence:

(23) EE (-21)A(V2)1 > — £(-z!)1erm(vty7 = 0, a > 0.

Thus, for a > 0, Jg > O and (a¥/30) > 0, as in the text (a= 1 implies
Jo = 0). For a< 0, we have not been able to sign Jo.

Finally, consider a change in the variability of debt expansion

(uy:
(24a) ul = a+ ozt; vl = (uty-!; a, scalar: €(z4) = 0.

(25a) (dv4/de) = -(vt)2. z).

As earlier, Y is determined by (13A); Jy < 0.

From (13A); :

(26) dq = EL(Z4) c(vty2em(zt, v2) 31

where M is defined by (22A). For a > 0:

(27A) sitt (fa) (7 <0

Hence:

(aga) ef (Z1) ¢(v1)2m3q < e(zt)E((vt)2m) = 0

and Jg < 0. Consequently, for a > 0, (dY/do) < 0 - i.e., an increase in

the variability of debt expansion causes an appreciation of the exchange
rate for that country.


27

real interest differential) if debt levels also change. Given the stock
of debt, the spot rate and forward premium move in opposite directions:
an appreciation of the currency is associated with an increase in the
forward premium and hence higher real returns on domestic (versus
foreign) bonds. However, if debt stocks change, there is no such
presumption. An increase in bond stocks (via a budgetary deficit) leads
to a depreciation of the spot and forward rate and an increase in the
forward premium and the relative real return on domestic bonds. Hence,
the effect of a budgetary deficit on spot rates and the forward premium
depends on how it is accomodated by monetary policy. A tight monetary
policy (decrease in the proportion of debt that is monetized) will lead
to higher relative real returns on domestic bonds; the spot rate may even
appreciate if the monetary policy is sufficiently restrictive.

The preceeding analysis presumed that the (distribution of)
rates of debt expansion were identical across countries. A natural
sequel is to ask how changes in the rate-or variability- of debt
expansion affects the spot and forward rates, and forward premium, 4
However, one must be careful in posing the question. A mean preserving
spread of wi (the rate of country 1 debt expansion) will increase the

expected real return (given nominal interest rates) on country 1 bonds

1
t

considering changes in the distribution of debt expansion (Upaq)> we

because of Jensen's inequality ((PE/P a1) = wy). Thus, rather than

consider changes in the distribution of the intertemporal price level (or

1 15

-1
(tay) )«

First, consider a change in policy that leaves the relative

variability of the intertemporal price ratio (compared to the rate of


comoarative static results concerning the rational expectations solution,
while the last section contains our conclusions and suggestions for

future research.

II) The Model

The purpose of our model is to allow us to derive asset demands
fron optimizing behavior, and to use these asset demands to determine how
exchange rates - and real interest rate differentials - are determined
under uncertainty. In specifying the model we have several objectives:
(i) to allow a relatively tractable general equilibrium solution to
emerge in the presence of uncertainty, and (ii) to specify a model so
that the conventional results predicted by the Monetary Theory of
Exchange Rate Determination will emerge if no uncertainty is present.
Thus, we wish to focus on how uncertainty affects exchange rates and real
interest rate differentials, not on how alternative specifications of
money demands may alter the exchange rate equation.

In order to accomplish these goals we use a relatively simple
consumption-loan model in which it is assumed that:

(i) there are two countries (i = 1, 2)

(ii) at any time, t, two generations are alive in each

country; members of generation t are born at t and die at

the end of (t + 1).

(iii) each generation in country i consist’ of N' individuals (a
constant).

(iv) there is one homogeneous and perishable good "produced" in
each country; aggregate output, Q'(t), and per capita
output, q'(t), is exogenous, and belongs to the "young" at

(v) each country, issues two nominal assets, currency (mi(t))
and bonds (B'(t)) denominated in its own currency unit.


14

where (ui, Ub 43) are assumed identically and independently distributed.
Similarly, (24> 245) have identical independent distributions, and ul is
distributed independently of ui, or a for any t. represents the
stationary values (of qi, ay g') about which K may vary 2l/

Under these assumptions, and using (30) - (32), the portfolio

rule (20) is given by:

11 gl yngged y-vyayyl 2 aly v2 yl ya.
(39) ELC #94 sy )agan Sea) ea) ea Meta) ear Mtea Me Meas) = 0

(39a) Vi = fu Ts ws OKO)

The interest rate, an endogenous variable, still appears in (39);
conceptually, it could be solved for in terms of the exogenous random
variables, though this is not analytically possible. The expectation in

1s taken over le exogenous variables (g ,A,q an which,
39) is tak th iabl "ats q') and y hich

t+l
in turn, depends on Keay and the distribution of these variables.

Since the exogenous (t+1) random variables in (39) have
stationary distributions, and since (hay) depends only on (t+1) events,

inspection of (39) indicates W, must be constant for all time.

* * -1
(40) W, = WY ¥ts Ye = COW") YK, I)
* -1
(41) eg = (Y,0E/0E) = COW*)YK ICY) (0d 702)


16

currency supplies. The other variables (except nominal prices) are
independent of the stock of debt, but depend on its composition. The
effect of transitory disturbances on the exchange rate can be seen from
(41); for the perfect foresight case, the impact of transitory
disturbances is seen from (7). A comparison indicates:

Proposition II: Transitory changes in money demand ai qi) have the same
proportionate effect on spot exchange rates under uncertainty and perfect
foresight. In either case, the forward rate (and, by assumption, future
spot rate) is unaltered.

The explanation for the equivalent effects is because
expectations concerning future exchange rates are unchanged. Note,
however, that the level of the exchange rate will not, in general, be the
same for the two cases. The transitory disturbances will, in either
case, affect the forward premium; under uncertainty, the expected real
interest differential will only be affected if it is initially non-zero.

If asset demand disturbances are transitory, while asset supply
disturbances (DF) are assumed permanent, the spot rate will be more
variable than the forward rate and the forward premium will be negatively
correlated with (independent of) the spot (forward) rate. Finally, the
forecast error, or "risk premium," embodied in the forward rate
(Elegy) - f,d/t,) will have a stationary distribution. The magnitude of
the risk premium cannot be ascertained without further assumptions.

As for transitory money demand disturbances, a transitory
change in money supply due to an open market operation will have the same
effect under uncertainty as under perfect foresight (for the latter case,
it does not matter whether the change in money supply is due to open

market operations or increased spending). From (41), holding o


Standard portfolio theory implies that agents will attempt to diversify
their portfolio, and hence implies that there is no ex ante reasori for
(expected) real returns to be equalized (unless agents are risk neutral).
Furthermore, recent empirical studies (e.g., Hansen and Hodrick ('980))
indicate that the forward rate is not an unbiased predictor of the future
spot rate, so that some risk premium seems to be embodied in the “orward
rate.

There have, of course, been other papers which analyze the
determinants of the forward premium. Driskill and McCafferty (1930a,
1982) analyze the determination of spot and forward rates in a model in
which ad hoc behavioral rules are postulated for the trade balance and
for the speculative demand for foreign exchange. Frankel (1979a)
analyzes the determinants of the forward premium in a portfolio setting
in which agents possess quadratic utility functions (or returns are
normally distributed). However, these models are essentially partial
equilibrium ones. It is the purpose of this paper to present a general
equilibrium model in which spot and forward rates (or interest rates) are
simultaneously determined, and in which the behavioral rules are derived
from individual optimizing behavior. Within this framework, we will
analyze how the forward premium is determined, and present some
comparative static results which show how changes in uncertainty, or
asset stocks, alter the spot and forward rates.

The plan of the paper is as follows. In section II we present
the basic model, and analyze the behavior of economic agents, given their
expectations about current and future prices. In section III we specify
the source of uncertainty, and show under what circumstances a rational

expectations solution can be derived. Section IV presents some


2A

Thus, for K = 1, Y = 1 solves (1A). Since Y is unique (given
distributions) and monotonically increasing in K; and since 0 < ay < 1:

(12A) Y gl as K 315 and (Y/K) glask al,
which is the result asserted in (69).
For changes in the distribution of vi;
(13A) oY, Ks @) = EL(V2(K/y)/¥ - viy(vey + wlyaly =o
(14A) yi = oz, @, a scalar.
Jy is given by (3A). Jo is:
(15A) [edo] = EL(-v!)(av! + (yv2(1 + W)/W))B%77 < 0, a > 0.

Hence, Y decreases as © increases, for a > 0; for a< 0, we have not been
able to sign (15A). Assume a >0 and define:

(16A) Y= -a90; ag > 0, a> 0.

Since Le(ed,) + ¥(Y0y)1 = 0, simplification of (3A) and (15A) yields:
(17a) eC (v4) {B01 : (agi) + (1 = a)(1 + ay)A}Bo"2] = 0.

oan par ner aa )] <0. Hence, (17A) implies (a < y).
(aga) et(vtyy(c(Ky)!/W2 - avtypo-1] = 0, where

(19A) C = (1 = a)(1 + ay) + WT - (ap/y))3 d = - a + (1 - a)ag + (a9/y)
Since C > 0, (18A) implies d >0, i.e.:

(20) y > a) > E(ay)/(1 + ¥(1 - a))] > 0, a > 0.

(Note (20) holds even for a < 0; however, a> cannot be signed in this
case.)

Next, consider a mean preserving spread of the reciprocal of
the rate of debt expansion:

(20A) visa+ oz; £(z}) = 0, E((z!)) > 0; a - scalar.

As earlier, Y is determined by (13A), and Jy <0. Consider Jos from
(13A):

(2A) dp = EL(-zt) pe-@tvev(7 + (1 = @)/W)) + av4]3]; let:
(22a) wivi, v2) = po-2cv2y(1 + ((1 - a)/W)) + av; and


(ji) its agents have a relatively greater demand parameter ah for
currency; or (iii) a larger fraction of its debt is issued in bonds. The
effect of country size is comparable to recent results from international
trade and capital movements (Grossman and Razin (1983)). Both effects
(i) and (ii) occur because they imply country 1 agents wish to hold
relatively more domestic currency which leads to an offsetting reduction
in demand for domestic bonds; hence, the relative return on these bonds
must rise. The third effect implies that relatively higher real returns
are required to induce agents to hold the larger (relative) supply of
country 1 bonds. Finally, note that the level of the debt does not
affect the forward premium or real interest differential since all debt
changes are assumed permanent, and interest rates are unaffected by
(balanced) changes in debt.

Next, consider the relation between the spot rate under
uncertainty and perfect foresight. Given the same (expected) rate of

monetary (debt) expansion, the perfect foresight exchange rate is:
* oe 1 2y) _ vend yn?

(61) a = (ME/ME)K((T + gt )/(1 + 9°) = K(DE/DE)

Under uncertainty:

~ venl yn? :
(62) = Y(D,/D,). Hence:

ee

v

* :
(63) 4 2, as Y 2K; i.e., as Kgl.

Proposition VI: Given identically, independently distributed rates of


Exchange Rate Determination and Real
Interest Differentials Under Uncertainty
by

Harvey E. Lapan

The extent to which economic agents consider domestic and
foreign assets to be perfect substitutes is of fundamental importance to
the conduct of monetary and fiscal policy. Under fixed exchange rates,
it is well known that if assets are considered perfect substitutes, then
monetary policy will be impotent for a small country, whereas for a
larger country monetary policy will have (essentially) the same effect in
all countries, regardless of which country initiates the change in
policy. Comparably, under a flexible exchange rate regime, sterilized
intervention to stabilize (or, simply, alter) exchange rates will be
useless, since asset holders view the foreign and domestic assets as
being equivalent. Furthermore, government deficits (due, for example, to
real absorption by the government) will leave exchange rates (and forward
rates) unaltered, provided none of the deficit is monetized.

In a world in which exchange rate movements are often large,
anc unpredictable, the assumption that agents consider assets denominated
in different currencies to be perfect substitutes is a strong one. Under
perfect foresight, and abstracting from transaction costs in switching
between assets, it is highly plausible that the real return on assets
should be equalized. However, under uncertainty, agents do not know,

can only know the true probability distribution of these returns.


Footnotes

1/ The Kareken-Wallace results imply one set of solutions is a constant,
but indeterminate, exchange rate. Under uncertainty, other solutions may
exist in which the exchange rate varies over time.

2/ Alternatively, the money demand functions could be "derived" by
introducing currency i real money balances in the utility function of
agents of country i. This procedure yields results that are comparable
to those discussed in the paper. We use the money demand functions (1)
to make our results comparable with the traditional monetary approach.
Under certainty, and "perfect capital" mobility, the exchange rate will
be independent of bond stocks.

3/ The use of a transactions technology is common in this type of model ;
for example, Helpman and Razin (1982a, 1982b) assume goods must be paid
for in the seller's currency. However, since their demand functions are
insensitive to interest rates, expectations do not affect the current
spot rate. In order to emphasize the role of expectations, we assume
that money demand is interest sensitive.

4/ Risk neutrality (a = 1) implies (Ci, St,,) are perfect substitutes
and, in general, leads.to corner soluttons. An alternative
specification: U = (ch)*(St 44) allows for interior solutions and risk
neutrality with respect to second period real income (a = 1). These two
specifications yield equivalent portfolio demand rules.

5/ Allowing the young to pay taxes would not alter the analysis. The
assumption that the nominal taxes due are known does simplify the
analysis. Since these taxes will be identified later with interest due
on the current debt, the assumption is consistent.

6/ Since short bond positions are allowed and since the Inada derivative
conditions hold, an interior solution will exist.

J/ These four equations are independent since the total demand for
wealth is endogenous. Alternatively, one bond market equation could be
replaced by the commodity market equilibrium equation.

8/ If one country could issue bonds denominated in the other country's

currency, it would become necessary to consider default risks. Clearly,
economic agents do not consider dollar denominated bonds issued by Brazil
to be perfect substitutes for those issued by the U.S.

9/ Although in our model bonds are outside wealth, an increase in the
Stock of bonds does not make agents feel richer because: (i) they are not
used as transfers; and (ii) agents assume the interest liability on these
bonds, Indeed, an increase in bonds (due to an increase in G) will cause
agents to reduce their consumption demand by (rPG), just as they would if
the spending were financed by taxes and agents attempted to equalize
consumption over their infinite lives (or, those of their offspring).


28

debt expansion) unchanged. Let:

1

(64) vi = oz!; yt

= (ot); E(z1) > 0

where © is a parameter, and zi a random variable. An increase in 0
corresponds to an increase in the expected real return, and its
variability, of country 1 bonds (given nominal interest rates);
equivalently, it corresponds to a decrease in the rate, and variability,
of nominal debt expansion.

Using (51), the impact of changes in © on Y (the portfolio

rule) can be found. As shown in the Appendix:
(65) Y= (-a,)0; y> ay > (ay)(1 + y (1 ~ a)?
Similarly, the "risk premium" on country 1 assets:
(66) k = ELfy/egyy)1 = (vk)! er (ves /v2.,)I

(67) k on - (a,/y)]

And the change in the expected real interest differential between country

1 and country 2 bonds is given by (59):
(68) dE E(a4)T = (1 +?) eLv'y ocysKy'/¥ (1 - (ap/y))

For risk neutrality (a= 1), ary and there is no change in the risk.


12

1 1 . 2 2
(32) (M, + BuYe = ea, (My + By)

1, gl = A ply! :
(33) (M, + BAC +A + ALY.) = APL O.. where:

-1
(lea), oT 5 pylal 4 y2n2
Tox, 1 3 = (Ng + Nae)

(34) AL
Equation (32) is an aggregate bond market equation, and (33) is aggregate
commodity demand; (34) is definitional. These four equations, plus (20)
-which determines the portfolio rule -determine re ep the ry, given
current asset stocks, output levels, and expectations. In the next
section we specify how a rational expectations equilibrium is

determined.

III) A Rational Expectations Solution

In order for the agent's expectations to be rational, these
expectations must be derived from the same structural model used to
derive the current period equilibrium. Unfortunately it is not
generally possible to find a closed form solution, 2? The principal
difficulty arises because it is not possible to obtain an analytic
solution for interest rates, regardless of expectations. Thus, if Pha

depends on ( ), it is not possible to specify the true distributior of

ri
t+l
prices.


15

In general, w depends on * and the distributions of the random
variables. Thus, one cannot infer from (41) how permanent changes affect
the exchange rate, but it can be used to analyze transitory
disturbances.

In addition, the forward rate, forward premium (fi /e,)> and

real interest differentials are:
1 2 *)eL 2
(42) f= Leg (1) + rg)/(1 + re)] = QW EO, /DE

* -1
(43) (Fle) = C0 + rh ys(1 + ry] = BR TY)

(44) a, = C01 + od) = (1 + 02)] = £1 + reyPe/OEIC(W*DE Phy) ~ (egy/Pe 4)

In (44), 4 is the (distribution of) real interest rate differentials between

assets. Due to our assumptions, pl 41 is linear homogeneous in of, and hence

(Db /P 41) is independent of time t events.

Given our assumptions, several results are immediately

apparent.

Proposition I: The spot and forward rates are linear homogenous in total

country 1 debt. A proportional expansion of both country i's bonds and money
. * i,

leads to an equiproportionate change in spot and forward rates, and P,; all

other variables are unaltered.

Thus, in general, the spot and forward exchange rates depend

upon the total stock of country i debt, and its composition, not just on