1. Suppose equations (9.1) and (9.2) are modified as follows:
yt= −αit+ ut
mt= −cit+ yt+ vt
where ut= ρuut−1+ϕt, vt= ρvvt−1+ψtand ϕ and ψ are white noise pro-cesses (assume all shocks can be observed with a one period lag). Assume the central bank’s loss function isE[y]2.
(a) Under a money supply operatingprocedure, derive the value of mt that minimizesE[y]2.
(b) Under an interest rate operatingprocedure, derive the value of it that minimizesE[y]2.
(c) Explain why your answers in (a) and (b) depend on ρu and ρv. (d) Does the choice between a money supply procedure and an interest
rate procedure depend on the ρis? Explain.
(e) Suppose the central bank sets its instrument for two periods (for example, mt = mt+1 = m∗) to minimize E[yt]2+ βE[yt+1]2 where 0 < β < 1. How is the instrument choice problem affected by the ρis?
a) Under a money supply procedure, the money demand relationship implies that the interest rate is it= c−1(yt+ vt− mt). Output is then equal to
yt = −α
c (yt+ vt− mt) + ut (116)
= α(mt− vt) + cut
c+ α
The objective is to pick mt to minimize E[y]2. The first order condition is 2α
c+ αE
α(mt− vt) + cut
c+ α
= 0
Since the shocks are assumed to be observed with a one period lag, E(vt) = ρvvt−1and E(ut) = ρuut−1,so this first order condition requires that mtsatisfy
αmt− αρvvt−1+ cρuut−1= 0 or
mt= ρvvt−1−c α
ρuut−1
The money supply is adjusted to offset the forecasted effects of vt and ut on output:
yt=cϕt− αψt
c+ α (117)
b) Under an interest rate procedure, output is equal to
yt= −αit+ ut (118)
and itis chosen to minimize E[y]2= E[−αit+ ut]2. The first order condition is
−2αE[−αit+ ut] = 0 or
it=ρuut−1 α
The interest rate is adjusted to offset the predicted aggregate demand shock, while money demand shocks do not affect output and so do not require any adjustment in the interest rate instrument.
c) As noted under parts (a) and (b), the optimal policy will involve trying to insulate output from the two shocks ut and vt. If these could be observed before policy is set, the optimal policies would be mt = vt−c
α
ut under a money procedure and it = 1
α
ut under an interest rate procedure. Under certainty equivalence (which holds in the linear model with a quadratic objective function), the optimal policy simply replaces utand vtwith the best forecast of the shocks, ρuut−1 and ρvvt−1.
d) The loss function under the m procedure is
L(m) = E
α
ρvvt−1−c
α
ρuut−1− vt
+ cut
c+ α
2
= E
c(ut− ρuut−1) − α(vt− ρvvt−1) c+ α
2
= E
cϕt− αψt c+ α
2
which is independent of both ρu and ρv. Under the interest rate procedure, the loss is
L(i) = E [ut− ρuut−1]2= E [ϕt]2
which is also independent of ρu and ρv. Consequently, the comparison between a money procedure and an interest rate procedure will not depend on either ρu
or ρv. Since the predictable component of the shocks (the component that does depend on ρu and ρv) is offset under both policies, the comparison will only depend on how well the different policies insulate output from the unforecastable shocks ( ϕtand ψt).
e) If the objective is to set m or i at time t for two period to minimize E[yt]2+ βE[yt+1]2, the analysis becomes more complicated and the comparisons between the money supply and the interest rate policies will depend on the serial correlation properties of the shocks. Starting with the money supply procedure, we can use (116) for output to write the loss function as
E
since, by assumption, m is fixed for two periods. The first order condition is 2α order condition, the optimal m∗ must satisfy
α(m∗− ρvvt−1) + cρuut−1+ β
Since m is fixed for two periods, it adjusts to offset what amounts to the average discounted expected shocks over the two periods. As a consequence, output will not be perfectly insulated from the forecasted components of ut, ut+1, vt, or vt+1: (which should be compared with equation 117) and
yt+1 = c
or
yt+1=c
ϕt+1+ ρuϕt
− α
ψt+1+ ρvψt
c+ α −
c(1 − ρu) ρuut−1− α (1 − ρv) ρvvt−1 (1 + β) (c + α)
Forecast errors made in period t, and therefore not fully offset, continue to affect output in period t+ 1 if the disturbances are serially correlated ( ρuϕt and ρvψt show up in the expressions for yt+1).
Under an interest rate policy, the objective is to pick i∗ to minimize E [−αi∗+ ut]2+ βE [−αi∗+ ut+1]2
so the first order condition is
−2α {E [−αi∗+ ut] + βE [−αi∗+ ut+1]} = 0
−αi∗+ ρuut−1− αβi∗+ βρ2uut+1= 0 or
i∗= (1 + βρu)ρuut−1 α(1 + β) and output in the two periods will equal
yt= ut−
1 + βρu
1 + β
ρuut−1= ϕt+β(1 − ρu) 1 + β ρuut−1 and
yt+1 = ut+1−(1 + βρu)ρuut−1 (1 + β)
= (1 + β)
ρ2uut−1+ ρuϕt+ ϕt+1
− (1 + βρu)ρuut−1 (1 + β)
= ϕt+1+ ρuϕt−(1 − ρu) (1 + β)ρuut−1
Since the variance of output under the two policies will now depend on ρu and ρu, the comparison of the loss functions under the two policies will no longer be independent of the serial correlation properties of ut and vt. For example, suppose ρu= 0 but ρv = 0. Under an interest rate policy, output does not depend on the v disturbances, so yt= ϕt and yt+1= ϕt+1, but under the money supply rule, equation (119) becomes
yt= cϕt− αψt
c+ α −
β
1 + β
α c+ α
(1 − ρv)ρvvt−1
and a money supply procedure is
E[yt]2m = while under an interest rate procedure,
E[yt]2i = σ2ϕ
2. Solve for the δis appearingin (9.11) and show that the optimal rule for the base is the same as that implies by the value of µ∗ given in (9.10).
The δis that appear in equation (9.11) are obtained by calculating the least squares forecast of each shock based on the observed value of the interest rate.
For the model of section 9.3.2, the equilibrium expression for the interest rate is given by equation (9.9):
i= v− ω + u α+ c + µ + h
We can use this to calculate the forecasts of v, ω, and u, conditional on ob-serving i. In the text, these forecasts are denoted ˆv, ˆω, and ˆu (see page 394).
From the least squares formula, the forecast of a variable y, conditional on x, is ˆy =
σx,y
σ2x
x where σx,y is the covariance between x and y and σ2x is the variance of x. (This assumes both y and x have zero means.)
Ap p lying this formula, we have ˆv =
Similarly,
The next stepis to substitute these expressions into the policy rule (9.11):
b =
Solving this for µ yields
µ= − (c + h) + α
σ2v+ ασ2ω σ2u
which proves that the policy rule (9.11) yields the same response to the interest rate as was found in equation (9.9), and the optimal policy rule is
b=
Show how the choice of an interest rate versus a money supply operating procedure depends on c2. Explain why the choice depends on c2.
Using the basic model given by equation (9.1) with the money demand equa-tion specified in the problem, interest rates and output under a money supply procedure are given by
and the loss function E(y)2 is minimized when
In contrast, under an interest rate procedure, y = −αi + u, the variance of output is minimized if i= 0, and the loss function then takes on the value
L(i) = σ2u
An interest rate rule is preferred if L(i) < L(m), or if σ2v >
which should be compared with equation (9.6) on page 390 of the text.
A large value of c2 (a large income elasticity of money demand) makes it more likely that a money supply procedure will be preferred. Consider the impact on output of a positive u shock. If c2 is large, the resulting rise in output has a large impact on money demand. This in turn causes interest rates to rise, offsetting the original rise in output. Thus, u shocks have a smaller impact on output under an m procedure when c2 is large. Similarly, a positive v that increases money demand and raises interest rates under an m procedure, will lower output but the decline in y has, when c2 is large, a strong impact in lowering money demand. As a result, interest rates need to rise less to maintain money market equilibrium after the positive v shock, and, with a smaller rise in i, y falls less.
These results can be illustrated by Figure 1. The negatively sloped solid line is the IS equation y = −αi when u takes on its expected value of 0. The positively sloped lines AA and BB give money market equilibrium when m= 0 and v takes on its expected value of 0 (that is, these lines show i =
c2
c1
y).
The line BB is drawn for a larger value of c2. The dotted negatively sloped line shows the results of a positive u shock; output rises less when c2is large; output rises from he the level associated with p oint C to D if c2 is small, but it rises only to E when c2 is larger. A positive v shock shifts AA and BB to AA and BB (since i=
1 c1
(c2y+ v), the vertical shift is the same for both). Again, output is less affected when c2 is large, falling only from C to F when c2 is large, rather than C to G.
Figure 5: Chapter 9, Problem 3 — The impact of c2 under a money supply operatingprocedure
Output
InterestRate
C
A
A A'
A'
B B'
B B'
D E F
G
4. Prices and aggregate supply shocks can be added to Poole’s analysis by usingthe followingmodel:
yt= yn+ a(πt− Et−1πt) + et (120)
yt= yn− α (it− Etπt+1) + ut (121)
mt− pt= β0− βit+ yt+ vt (122) Assume the central bank’s objective is to minimize E
λy2+ π2
, and that are disturbances are mean zero, white noise processes. Both Et−1πt and the policy instrument must be set prior to observingthe current values of the disturbances.
(a) Calculate the expected loss function if itis used as the policy instru-ment. (Hint: Give the objective function, the instrument will always be set to ensure expected inflation is equal to zero.)
(b) Calculate the expected loss function if mt is used as the policy in-strument.
(c) How does the instrument choice comparison depend on
i. the relative variances of the aggregate supply, demand, and money demand disturbances?
ii. the weight on stabilizing output fluctuations λ?
Note: There is a typo in this problem; the loss function should be E
λ(y − yn)2+ π2 (or one can simply assume yn= 0 as a normalization).
a) Under an interest rate policy, the money demand equation given by (122) is not needed. Using the “hint” and setting Et−1πt = Etπt+1 = 0, equation (140) implies yt− yn = −αit+ ut. Using this in (120), inflation will equal πt= 1a(−αit+ ut− et). This means we can write the policy problem in terms of the policy instrument it as
minit E
λ(−αit+ ut)2+ 1
a(−αit+ ut− et) 2
The first order condition is
−2αE
λ(−αit+ ut) +1
a(−αit+ ut− et)
= 0 (123)
The problem specified that the policy instrument must be set before observing the disturbances, so in evaluating the first order condition, E(ut) = E(ut) = E(et) = 0. Thus, (100) becomes −αλit− αa1it= 0 or
it= 0 With this setting for the nominal interest rate,
yt− yn= −αit+ ut= ut (124) and
πt= 1
a(−αit+ ut− et) =ut− et
a
Notice that under a policy that sets i = 0, inflation is a mean zero, serially uncorrelated process, so Et−1πt= Etπt+1= 0 as was assumed.
Using these results, the expected loss under an interest rate policy, L(i) =(1 + a2λ)σ2u+ σ2e
a2 (125)
b) Under a money rule, we need to use equation (122), solving it for the nominal interest rate and then use this result to eliminate it from equations (120) and (121). Since equations (120) and (140) are expressed in terms of
the rate of inflation while (122) involves the p rice level, we can either exp ress inflation as pt− pt−1, and then solve for the price level, or, since pt−1 is known when policy is set, we could replace mt− pt in (122) with mt+ pt−1− πtand solve for the rate of inflation. Since the loss function is expressed in terms of inflation, this latter approach is more convenient.
From (122), the nominal rate of interest is
it=c0+ πt− mt+ pt−1+ yt+ vt
c (126)
Substituting this into (140), and using the earlier hint about expected inflation, we have
yt− yn = −α
c0+ πt− mt+ pt−1+ yt+ vt
c
+ ut
= α(mt− c0− πt− pt−1− yn− vt) + cut
c+ α which can be solved jointly with (120) to yield
yt− yn=aα(mt− c0− pt−1− yn− vt) + acut+ αet
a(c + α) + α (127)
πt= α(mt− c0− pt−1− yn− vt) + cut− (c + α) et
a(c + α) + α (128)
Now substitute these two solutions into the loss function. The policy problem is then
minµt E
λ
aα(µt− vt) + acut+ αet
a(c + α) + α
2 +
α(µt− vt) + cut− (c + α) et
a(c + α) + α
2
where, for convenience, µt has been defined as mt− c0− pt−1− ynand can be viewed as the policy instrument. The first order condition for the choice of µt
is
2α a(c + α) + α
aλ
aαµt a(c + α) + α
+
αµt a(c + α) + α
= 0
where we have used the fact that at the time policy is chosen, Eut = Eet = Evt= 0. The first order condition is satisfied for µt= 0, or
mt= c0+ pt−1+ yn
Using (127) and (128), output and inflation under a money instrument are yt− yn=acut− aαvt+ αet
a(c + α) + α
and
πt= cut− αvt− (c + α) et
a(c + α) + α and the loss function is
L(m) =
1 + a2λ
c2σ2u+ α2σ2v +
α2λ+ (c + α)2 σ2e
[a (c + α) + α]2 (129)
c) The instrument choice hinges on a comparison of the loss L(i) given in (125) and the loss L(m) given in (129), with an interest rate instrument preferred if
L(i) < L(m) or if
(1 + a2λ)σ2u+ σ2e a2 <
1 + a2λ
c2σ2u+ α2σ2v +
α2λ+ (c + α)2 σ2e
[a (c + α) + α]2 (130)
This can be rewritten with some rearranging as implying L(i) < L(m) if
σ2v>
(a (c + α) + α)2− a2c2
a2α2 σ2u+
2a (c + α) + α(1 − a2λ) (1 + a2λ) σ2e
This shows that the comparison depends on the different variance terms. A money oriented operating procedure is less likely to be desirable if money demand shocks are large (i.e., σ2v is large). The coefficient on σ2u is positive, so an interest rate rule is more likely to be preferred if aggregate demand shocks are important (i.e., σ2u is large), while it is also likely to be preferred if supply disturbances are large (i.e., σ2e is large). Notice that if σ2e is zero (no supply shocks), the comparison is independent of the preference weight λ.
The weight on stabilizing output fluctuations, λ, affects the comparison only if σ2e > 0. In the absence of aggregate supply shocks, there is no conflict in this model between stabilizing output and stabilizing inflation, so the operating procedure comparison will be independent of λ. When aggregate supply shocks are present ( σ2e>0), then there can be conflicts between stabilizing output and stabilizing inflation. If output objectives are very important ( λ large), then it is more likely an interest rate procedure will be preferred. To understand why, consider what happens in the face of a positive aggregate supply shock. Under an interest rate procedure, aggregate demand remains constant (see equation 124), so output is stabilized and inflation must fall. Such a policy will be preferred if λ is large. Under a money supply procedure, in comparison, output will rise and inflation will fall. Since both adjust, inflation falls by less than under the i policy. A policy maker who cares more about inflation stabilization (i.e., has a lower λ) will prefer money supply operating procedure.
5. Using the intermediate target model of section (9.3.3) and the loss function (9.15), rank the policies that set itequal toˆıt, iTt, andˆιt+ µ∗xt.
The basic model of section 9.3.3 consists of the following equations:
yt= a(πt− Et−1πt) + zt
yt= −α (it− Etπt+1) + ut
mt− pt−1− πt= yt− cit+ vt
These appeared as equations (9.12) - (9.14) on page 397. The loss function (9.15) is
V = E (π − π∗)2 The first policy to evaluate sets
it= ˆıt= π∗+
1 α
(ρuut−1− ρzzt−1)
(see equation 9.17, page 397). The value of the loss function under this policy was given at the bottom of page 397:
V(ˆıt) =
1 a
2
σ2ϕ+ σ2e
(131)
Under the second policy,
it= iTt = ˆıt+(1 + a)ϕt− et+ aψt ac+ α(1 + a)
(see equation 9.21 on page 399). The inflation rate is (see page 399) πt
iTt
= π∗+cϕt− (α + c)et− αψt ac+ α(1 + a)
and the value of the loss function under this policy was given in the middle of page 400:
V(iTt) =
1
ac+ α(1 + a)
2
c2σ2ϕ+ (α + c)2σ2e+ α2σ2ψ
(132) Comparing (131) and (132),
V(ˆıt) − V (iTt) =
1 a
2
σ2ϕ+ σ2e
−
1
ac+ α(1 + a)
2
c2σ2ϕ+ (α + c)2σ2e+ α2σ2ψ
=
2acα(1 + a) + α2(1 + a)2
σ2ϕ+ 2αa2(c + α)σ2e a2(ac + α(1 + a))2
−
1
ac+ α(1 + a)
2
α2σ2ψ
The first term is positive, indicating that the intermediate targeting rule leads to a smaller loss ( V(ˆıt) > V (iTt)) if the only disturbances are demand and supply shocks ( ϕ and e). As discussed in the text, however, the intermediate targeting procedure can do worse if money demand shocks are important ( V(ˆıt) < V (iTt)
Using the definition of ˆιt, the rate of interest under this policy is it= ˆιt+ µ∗xt= π∗+
To evaluate the loss function under this policy, use equation (9.16) of the text to find the equilibrium rate of inflation when it= ˆιt+ µ∗xt:
Problem 2 showed that the value of µ∗was related to the best forecasts of ϕ and e, conditional on observing x. We can write inflation under this policy as
π(ˆιt+ µ∗xt) = π∗+ it is clear that the variance of inflation around π∗ will be smaller with the ˆιt+ µ∗xt policy since the variance of [ϕt− E(ϕt− | x)] is less than or equal to the variance of ϕ, and similarly for the comparison of the variances of [et− E(et| x)] and et. Therefore,
V∗≡ V (ˆιt+ µ∗xt) ≤ V (ˆıt)
To compare the loss under the intermediate target policy iT and the policy that optimal uses information (ˆιt+ µ∗xt), use the equation near the bottom of page 400 to write
iTt = ˆιt+ µTxt
Usingequation (9.16). inflstion can be written as πt(iTt) = (a + α)π∗− α
ˆιt+ µTxt
+ ut− zt
a
= π∗+ϕt− et− αµTxt a
so the loss function V(iT) is equal to the variance of 1
a
ϕt− et− αµTxt . Inflation around π∗under theˆιt+ µ∗xtpolicy is
πt(ˆιt+ µ∗xt) = π∗+ϕt− et− αµTxt a and the loss function V∗ is the variance of1
a
(ϕt− et− αµ∗xt). But since µ∗ was chosen to minimize this variance, it must be that V∗≤ V (iT).
6. Show that if the nominal interest rate is set accordingto (9.17), the ex-pected value of the nominal money supply is equal tom given in (9.19).ˆ According to equation (9.17) on page 397, the interest rate takes the value
it= π∗+
1 α
(ρuut−1− ρzzt−1)
With inflation given by equation (9.18) and output by (9;12), the money demand equation (9.14) can be written as
mt− pt−1 = πt+ a (πt− π∗) + zt− cit+ vt
= π∗+ (1 + a)
ϕt− et
a
+ zt− cπ∗−c α
(ρuut−1− ρzzt−1) + vt
since Et−1πt = π∗. Taking exp ectations as of time t− 1 of this equation and solving for Et−1mt,
Et−1mt= (1 − c)π∗+ pt−1−c α
ρuut−1+ 1 + c
α
ρzzt−1+ ρvvt−1 which is the same as the expression for m in equation (9.19).ˆ
7. Suppose the central bank is concerned with minimizingthe expected value of a loss function of the form
L= E[T R]2+ χE[if]2
which depends on the variances of innovations to total reserves and the funds rate (χ is a positive parameter). Usingthe reserve market model of section 9.4.2, find the values of φdand φbthat minimize this loss function.
Are there conditions under which a pure nonborrowed reserves or a pure borrowed reserves operatingprocedures would be optimal?
The reserves market model from section 9.4.2 consists of a total reserves demand equation, a borrowed reserves demand equation, a supply of nonborrowed reserves equation, and an equilibrium condition. These are specified as
T R= −αif+ vd
BR= b(if− id) + vb
N BR= φdvd+ φbvb+ vs and the equilibrium condition that
T R= BR + NBR
= αif+ b(if− id) + (φd− 1)vd+ (1 + φb)vb+ vs
Substituting these first three equations into the equilibrium condition and solving for the funds rate if yield
if = first order conditions will be
−2αE[−αif+ vd]
To evaluate these, note that
∂if
φd= 1 −α(a + b)
α2+ χ (135)
Equation (134) yields
−αE
α(1 + φb)vb a+ b
−vb a+ b
+ χE
−(1 + φb)vb a+ b
−vb a+ b
= 0
or
−α2
1 + φb (a + b)2
σ2b+ χ
1 + φb (a + b)2
σ2b = 0
which can be solved for the optimal φb, yielding
φb = −1 (136)
To understand these results, start first with the φb = −1 finding. A shock to borrowed reserve demand should generate an equal but opposite movement in nonborrowed reserves. This keeps total reserves unchanged. Since neither total reserve supply or demand have changed, the funds rate is left unchanged.
Thus, setting φb = −1 and accommodating shifts in borrowed reserve demand completely insulates both T R and if from vb shocks.
From (135), the optimal value of φdis equal to 1 −α(a+b)α2+χ is less than1 and depends on the preference parameter χ. In response to a shock to total reserve demand, reserve supply adjusts to fully accommodate the shift if φd = 1; this would succeed in insulating the funds rate from the shock, but it would lead total reserves to move one-for-one with vd. By setting φd <1, reserve supply less than fully accommodates the shift in reserve demand. This means the funds rate rises (falls) if vd >0 ( < 0). This means the funds rate moves more, but total reserves move less, and this will be optimal since the policy maker cares about both E[T R]2 and E[if]2