solution manual chapter 3.pdf
TRANSCRIPT
Monetary Theory and Policy:
Problem Solutions∗
Carl E. Walsh
University of California, Santa Cruz
March 16, 1999
Contents
1 Chapter 2: Money in a General Equilibrium Framework 2
2 Chapter 3: Money and Transactions 8
3 Chapter 4: Money and Public Finance 20
4 Chapter 5: Money and Output in the Short Run 27
5 Chapter 6: Money and the Open Economy 43
6 Chapter 8: Discretionary Policy and Time Inconsistency 59
7 Chapter 9: Monetary-Policy Operating Procedures 81
8 Chapter 10: Interest Rates and Monetary Policy 97
9 Typos 109
∗Solutions c© 1998 by Carl E. Walsh. Comments and corrections should be sent to [email protected].
1
1 Chapter 2: Money in a General Equilibrium
Framework
1. Calvo and Leiderman (1992): A commonly used specification of the de-mand for money, originally due to Cagen (1956), assumes m = Ae−αit
where A and α are parameters and i is the nominal rate of interest. Inthe Sidrauski (1967) model, assume that utility is separable in consump-tion and real money balances: u(ct,mt) = w(ct) + v(mt), and furtherassume that v(mt) = mt (B −D lnmt) where B and D are positive pa-rameters. Show that the demand for money is given by mt = Ae−αtit
where A = e(BD−1)and αt = w′(ct)/D.
The basic condition from which one can derive the demand for money inSidrauski’s money-in-the-utility function model is given by equation (2.23) onpage 57. This equation states that the ratio of the marginal utility of money tothe marginal utility of consumption depends on the nominal rate of interest:
um(ct,mt)
uc(ct,mt)=
it1 + it
≈ it
Notice that the expression has been simplified by employing the approximationx/(1 + x) ≈ x for small x. Sidrauski developed his model in continuous time,in which case the first order condition takes the exact form
um(ct,mt)
uc(ct,mt)= it
Using the proposed utility function, um = B −D −D lnmt and uc = w′(ct), sothis condition becomes
umuc
=B −D −D lnmt
w′(ct)=B/D − 1− lnmt
w′(ct)/D= it
Rearranging yields
lnmt = (B
D− 1)− w′(ct)it
D
or
mt = e(BD−1)e−
w′(ct)D
it
2. Suppose u(ct,mt) =∑∞
i=0 βi [ln ct +mte
−γmt ], γ > 0 and β = 0.95. As-sume the production function is f(k) = k0.5 and δ = 0.02. What rate ofinflation maximizes steady-state welfare? How do real money balances atthe welfare maximizing inflation rate depend on γ?
2
The steady-state welfare maximizing nominal rate of interest is iss = 0 (seesection 2.3.1.2, pages 61-64) at which point um = 0. If R is the gross real rateof interest (one plus the real rate of interest), 1 + i = R(1 + π) and the rate ofinflation that yields a zero nominal rate of inflation is
πss =1
R− 1
In the steady-state, R is equal to 1/β, or 1/R = β = 0.95 (see equation 2.19,page 54). Hence, the optimal rate of inflation is 0.95− 1 = −0.5 or a 5% rateof deflation.
To determine how money demand depends on the parameter γ, use the rep-resentative agent’s first order condition (see equation 2.23, page 57), evaluatedat the steady-state nominal rate of interest:
umuc
=iss
1 + iss
Given the form of the utility function, this becomes
umuc
= ct (1− γmt) e−γmt =
iss
1 + iss
At the welfare maximizing inflation rate, iss = 0, which requires 1− γmss = 0or
mss =1
γ
Thus, real money demand is decreasing in γ.
3. Assume that mt = Ae−αit where A and α are constants. Calculate thewelfare cost of inflation in terms of A and α, expressed as a percentage ofsteady-state consumption (normalized to equal 1). Does the cost increaseor decrease with α? Explain why.
A traditional method for determining the welfare cost of inflation involvescalculating the area under the money demand function. That is, the loss inconsumer surplus when the interest rate is equal to i > 0 is given by
l(i, A, α) ≡∫ i
0
Ae−αxdx
Evaluating this integral yields
l(i, A, α) =A
α
(1− e−αi
)(1)
3
as the welfare cost of an inflation rate of π = i−r if r is the real rate of return.The effect of α on this cost is
∂l(i, A, α)
∂α=αA(ie−αi)−A(1− e−αi)
α2=
(A
α2
)[e−αi(1 + αi)− 1
] ≤ 0
The sign depends on e−αi(1 + αi) − 1, but this is always negative ( e−x(1 + x)is maximized when x = 0 at which point e−x(1 + x) = 1; it then declines withx).
From the specification of the money demand equation, the interest elasticityof money demand is −αi, so money demand is more sensitive to the interestrate the larger is α. As the nominal interest rate rises with an increase in infla-tion, households respond by reducing their demand for money, thereby helpingto reduce the distortion generated by the inflation tax. The greater the interestsensitive of money demand, the lower will be the welfare cost of the inflationtax.
4. Suppose a nominal interest rate of im is paid on money balances. Thesepayments are financed by a combination of lump-sum taxes and printingmoney. Let a be the fraction financed by lump-sum taxes. The govern-ment’s budget identity is τ t+ vt = immt, with τ t = aimmt and vt = θmt.Using Sidrauski’s model,
(a) Show that the ratio of the marginal utility of money to the marginalutility of consumption will equal r + π − im = i− im. Explain why.
(b) Show how i−im is affected by the method used to finance the interestpayments on money. Explain the economics behind your result.
(a) The budget constraint in the basic Sidrauski model must be modified totake into account the interest payments on money and that net transfers ( τ inequation (2.12), page 52) consists of two components, the first being the lump-sum transfer v and the second being the lump-sum tax τ . Thus, the budgetconstraint becomes
f(kt−1) + (1− δ)kt−1 +1 + im
1 + πmt−1 − τ t + vt = ct + kt +mt
where population growth has been ignored for simplicity. The value function forthe problem is still given by (2.13), but the first order conditions change becauseof the change in the budget constraint. In particular, equation (2.15) becomes
um − β [fk + 1− δ]Vω(ωt+1) +β(1 + im)
1 + πVω(ωt+1) = 0 (2)
The first order condition for consumption (see 2.14), together with the envelopecondition (see 2.17) implies
uc(ct,mt) = β [fk + 1− δ]Vω(ωt+1) = β [fk + 1− δ]uc(ct+1,mt+1)
= βRuc(ct+1,mt+1)
4
Using this result, equation (2) can be rearranged, resulting in
umuc
= 1−[β(1 + im)
1 + π
]uc(ct+1,mt+1)
uc(ct,mt)= 1−
[β(1 + im)
1 + π
](1
βR
)
= 1− β(1 + im)
R(1 + π)=R(1 + π)− (1 + im)
R(1 + π)=i− im
1 + i
The ratio of the marginal utility of money to consumption is set equal to theopportunity cost of money. Since money now pays a nominal rate of interestim, this opportunity cost is i− im, the difference between the nominal return oncapital and the nominal return on money.
(b) From the government’s budget constraint, interest payments not financedthrough lump-sum taxes must be financed by printing more money. Hence, v =θm = (1 − a)imm, or the rate of money growth will equal θ = (1 − a)im. Inthe steady-state, π = θ. This means that π = (1−a)im. Hence, the opportunitycost of money is given by
i− im ≈ r + π − im = r + (1− a)im − im = r − aim
where r = R − 1. Paying interest on money affects the opportunity cost ofmoney only if a > 0. Printing money to finance interest payments on moneyonly results in inflation; this raises the nominal interest rate i, thereby offsettingthe effect of paying interest.
5. Assume u(c,m) = −c−a[1 + (m∗ −m)2
], a > 0. Normalize so that the
steady-state value of consumption is equal to 1 (css = 1). Using equation(2.23) of the text, show that there exist two steady state equilibrium valuesfor real money balances if aIss < 1. (Recall that I = i/(1 + i) where i isthe nominal rate of interest.)
From equation (2.23),
um(css,mss)
uc(css,mss)= Iss
Using the utility function specified in the question,
um(c,m) = 2c−a(m∗ −m)
and
uc(c,m) = ac−(1+a)[1 + (m∗ −m)2
]Therefore
um(css,mss)
uc(css,mss)=
2(css)−a(m∗ −mss)
a(css)−(1+a) [1 + (m∗ −mss)2]= I (3)
5
We now need to solve this equation for mss. Let x ≡ (m∗−mss). Then equation(3) becomes
2(css)−ax = aI(css)−(1+a)[1 + x2
]If css is normalized to equal 1, this becomes
2x = aI[1 + x2
]which can be rewritten more explicitly as a quadratic in x:
x2 − 2
aIx+ 1 = 0
From the quadratic formula,
x =
2aI ±
√4
(aI)2 − 4
2=
1
aI±√
1
(aI)2− 1
There will be two real solutions if and only if
1
(aI)2− 1 > 0
which holds for
aI < 1
If this condition is satisfied, both solutions for x are positive (so that mss < m∗).This can be verified by noting that aI < 1 implies
1
aI−√
1
(aI)2− 1 > 0
6. In Sidrauski’s money-in-the-utility-function model augmented to includevariable labor supply, money is superneutral if the representative agent’spreferences are given by∑
βiu(ct+i,mt+i, lt+i) =∑
βi(ct+imt+i)bldt+i
but not if they are given by∑βiu(ct+i,mt+i, lt+i) =
∑βi(ct+i + kmt+i)
bldt+i
Discuss. (Assume output depends on capital and labor and the aggregateproduction function is Cobb-Douglas.)
6
The steady-state values of css, kss, lss, yss must satisfy the following fourequations (see pages 65-67):
uluc
= −fl (kss, 1− lss) (4)
fk(kss, 1− lss) =
1
β− 1 + δ (5)
css = f(kss, 1− lss)− δkss (6)
yss = f(kss, 1− lss) (7)
If the single period utility function is of the form (ct+imt+i)bldt+i, then
uluc
=dcbmbld−1
bcb−1mbld=dc
bl
is independent of m and equations (4) - (7) involve only the four unknowns css,kss, lss, yss. These can be solved for css, kss, lss, and yssindependently of mor inflation. Superneutrality holds.
If the utility function is (ct+i + kmt+i)bldt+i, then
uluc
=d(c+ km)bld−1
b(c+ km)b−1ld=d(c+ km)
al
which is not independent of m. Thus, equations (4) - (7) will involve 5 un-knowns ( css, kss, lss, yss, and mss) and cannot be solved independently of themoney demand condition and inflation. Superneutrality does not hold.
7. Suppose the representative agent does not treat τ t as a lump sum transfer,but instead assumes her transfer will be proportional to her own holdingsof money (since in equilibrium, τ is proportional to m). Solve for theagent’s demand for money. What is the welfare cost of inflation?
This question is basically the same as Question 4. If the transfer is viewed bythe individual as proportional to her own money holdings, then this is equivalentto the individual viewing money as paying a nominal rate of interest. If this isfinanced via lump-sum taxes, changes in inflation do not change the opportu-nity cost of holding money — a rise in inflation that depreciates the individual’smoney holdings is offset by the increase in the transfer the individual anticipatesreceiving.
7
2 Chapter 3: Money and Transactions
1. Suppose the production function for shopping takes the form ψ = c =ex(ns)amb, where a and b are both positive but less than 1 and x is a
productivity factor. The agent’s utility is given by v(c, l) = c1−Φ
1−Φ + l1−η
1−ηwhere l = 1− n− ns and n is time spend in market employment.
(a) Derive the transaction time function g(c,m) = ns.
(b) Derive the money in the utility function specification implied bythe shopping production function. How does the marginal utilityof money depend on the parameters a and b? How does it depend onx?
(c) Is the marginal utility of consumption increasing or decreasing in m?
(a) From the shopping production function,
g(c,m) = ns =( c
exmb
)1/a(8)
(b) From the definition of the agent’s utility,
v(c, l) =c1−Φ
1−Φ+
(1− n− ns)1−η
1− η
=c1−Φ
1−Φ+
(1− n− ( c
exmb
)1/a)1−η
1− η≡ u(c, n,m)
The marginal utility of money is
∂u(c, n,m)
∂m=
∂v(c, l)
∂l
(∂l
∂m
)
=(1− n− (ns)
1/a)−η
(bns
am
)(9)
The time spend shopping can be written as c1/ae−x/am−b/a. The marginal pro-ductivity of money in reducing shopping time is given by (b/a)(ns/m), so anincrease in b/a increases the effect additional money holdings have in reducingthe time needed for shopping. Additional money holdings result in more leisure(and more utility) when b/a is large, thus acting to increase the marginal utilityof money. In terms of equation (9), ∂l
∂m rises with b/a. But the marginal utility
of leisure a decreasing function of total leisure, so ∂v(c,l)∂l declines.
The effect of x on the marginal utility of money, for given c and n, operatesthrough ns and represents a productivity shift; an increase in x reduces the timeneeded for shopping for given values of c and m. This affects the productivityof m in the shopping time production function. The marginal product of moneyin reducing shopping time is (b/a)c1/ae−x/am−(1+b/a). This is decreasing in x;
8
a higher x decreases the marginal effect of m in reducing shopping time, somoney is less “productive.”
(c) An increase in consumption affects utility in two ways. First, consump-tion directly yields utility; vc > 0. This represents the effect of consumptionon utility, holding leisure constant. Since leisure is being held constant, vc isindependent of m. Second, higher consumption increases the time devoted toshopping, as this reduces the time available for leisure. This effect will dependon the level of money holding. From (8), consumption and money are comple-ments in producing shopping time, and higher money holdings reduce the effectof higher c on ns. This means that with higher money holdings, an increase inc has less of an effect in reducing leisure time and will therefore lead a rise inconsumption to have a larger overall positive effect on utility.
2. Define superneutrality. Carefully explain whether the Cooley-Hansen cash-in-advance model exhibits superneutrality. What role does the cash-in-advance constraint play in determining whether superneutrality holds?
A model exhibits the property of superneutrality if the real equilibrium (out-put, consumption, capital, etc.) is independent of the rate of nominal moneygrowth. Superneutrality normally is interpreted to refer to the steady-state equi-librium of a model. As demonstrated in Chapter 2, the Sidrauski model dis-plays superneutrality with respect to the steady-state, but changes in the in-flation rate will generally affect the short-run equilibrium. If the ratio of themarginal utilities of leisure and consumption is independent of money holdings,then Sidrauski’s model is superneutral in the short-run also (see pages 65-67).
The Cooley-Hansen model does not display superneutrality. Different ratesof inflation affect the opportunity cost of holding money. Through the cash-in-advance constraint, inflation affects the marginal cost of consumption sinceconsumption is treated as a cash good. Higher inflation induces a substitutionaway from cash goods and towards credit goods. In Cooley and Hansen’s model,leisure is a credit good; cash is not needed to purchase leisure. As a result,changes in the steady-state rate of inflation alter the demand for leisure and thesupply of labor. This was shown in equation (3.29) on page 110, where Θ wasequal in the steady-state to one plus the inflation rate.
3. Is the steady-state equilibrium in the Cooley-Hansen cash-in-advance modelaffected by any of the following modifications? Explain.
(a) Labor is supplied inelastically (normalize so that n = 1, where n isthe supply of labor).
(b) Purchases of capital are also subject to the cash-in-advance constraint(i.e. one needs money to purchase both consumption and investmentgoods);
9
(c) The growth rate of money follows the process ut = γut−1+ϕt where0 < γ < 1 and ϕ is a mean zero i.i.d. process.
(a) Following on the previous problem, one important modification when la-bor is supplied inelastically is that Cooley and Hansen’s model will now displaysuperneutrality. Without a labor-leisure choice, the model becomes essentiallythe model of section 3.3.1.
(b) Referring to the model of section 3.6.1, the cash-in-advance constraintwould become
Ptct + Pt [kt − (1− δ)kt−1] ≤Mt−1 + Tt−1
where kt − (1− δ)kt−1 is equal to net purchases of capital. Dividing by Pt, thisbecomes
ct + it ≤ mt−1
Πt+ τ t ≡ at (10)
where it is net investment (kt − (1− δ)kt−1).The value function for this problem is
V (at, kt−1) = max {u(ct, 1− nt) + βEtV (at+1, kt)}where at+1 = mt
Πt+1+ τ t+1, kt = f(kt−1, nt) + (1 − δ)kt−1 + at − ct −mt and
the maximization is subject to the cash in advance constraint (10). Let λ bethe Lagrangian multiplier associated with the budget constraint and let µ bethe Lagrangian multiplier associated with the cash-in-advance constraint. If weassume a standard Cobb-Douglas production function (yt = eztkαt−1n
1−αt ), then
the budget constraint is
eztkαt−1n1−αt + (1− δ)kt−1 + at ≥ ct + kt +mt
and the first order conditions for ct, kt, mt, and nt, together with the envelopeconditions, are
uc(ct, 1− nt)− λt − µt = 0 (11)
βEtVk (at+1, kt)− λt − µt = 0 (12)
βEt
(1
Πt+1
)Va(at+1, kt)− λt = 0 (13)
−un(ct, 1− nt) + (1− α)
(ytnt
)βEtVk(at+1, kt) = 0 (14)
Va(at, kt−1) = λt + µt (15)
10
Vk(at, kt−1) = λt
(α
ytkt−1
+ 1− δ
)+ µt(1− δ) (16)
The Lagrangian µ appears in this last condition because higher capital at thestart of the period reduces the cash needed to achieve a given value of kt; onlynet purchases (kt − (1− δ)kt−1) are subject to the cash-in-advance constraint.
Since (15) implies EtVa(at+1, kt) = Et
(λt+1 + µt+1
), equations (11) - (16)
can be used to derive the following conditions, which should be compared toequations (3.51), (3.52), and the two equations following (3.52) on page 126:
uc(ct, 1− nt) = λt + µt (17)
βEt
(λt+1 + µt+1
Πt+1
)= λt (18)
−un(ct, 1− nt) + (λt + µt) (1− α)
(ytnt
)= 0 (19)
λt + µt = βEtVk(at+1, kt) = βEt
[Rtλt+1 + µt+1(1− δ)
](20)
where Rt = αyt+1
kt+ 1 − δ. The first two equations are identical to (3.51) and
(3.52). The next two differ. According to (??), the marginal utility of leisure isset equal to the utility value of the marginal product of labor, but now accountmust be taken of the fact that any additional income requires cash to be spent.That is why the marginal product of labor is multiplied by λt + µt and not justλt. According the (20), the value of an additional purchase of capital (whichcosts λt + µt) is the additional future return (the Rtλt+1term) and the value ofrelaxing the future cash-in-advance constraint that comes from reducing futurenet purchases (the µt+1(1− δ) term).
Turning to an analysis of the steady-state, (18) implies that
µss = λss(Πss
β− 1
)
which implies Πss ≥ β will be requires for the existence of a steady-state sinceµ must be nonnegative. Now eliminate µ from the steady-state version of (20):
λss(Πss
β
)= β
[Rssλss + λss
(Πss
β− 1
)(1− δ)
]
or, recalling that Rss−1+δ is the steady-state marginal product of capital αyss
kss ,
αyss
kss=
(Πss
β
)(1
β− 1 + δ
)
which depends on the rate of inflation. Thus, superneutrality does not hold whencapital purchases are also subject to the cash-in-advance constraint. Notice that
11
this conclusion would hold even if labor is supplied inelastically as in part (a)of this question (see Problem 5 below). By imposing a tax on capital purchases,inflation affects the steady-state capital stock and kss is decreasing in Πss.
For a complete discussion of the implications of making the cash-in-advanceconstraint apply to both consumption and capital or only to consumption, seeAbel (1985).
(c) The steady-state depends on the average rate of money growth since thatpins down average inflation. It does not depend on the transitory dynamics ofthe monetary supply process, although the short-run dynamics will.
4. Money-in-the-utility-function and cash-in-advance constraints are alterna-tive means for constructing models in which money has positive value inequilibrium.
(a) What strengths and weaknesses do you see with each of these ap-proaches?
(b) Suppose you wanted to study the effects of the growth of credit cardson money demand. Which approach would you adopt? Why?
Both the money-in-the-utility function approach and the cash-in-advance ap-proach are best viewed as convenient short-cuts for generating a role for money.If we believe that the major role money plays is to facilitate transactions, thenin some ways the CIA approach has an advantage in making this transactionsrole more explicit. It forces one to think more about the exact nature of thetransactions technology and the timing of payments (e.g., can current period in-come be used to purchase current period consumption?). On the other hand, therather rigid restrictions the CIA typically places on transactions are certainlyunattractive. In modern economies we normally have multiple means that can beused to facilitate the transactions we undertake. Also, the generally exogenousdistinction between cash and credit goods is troublesome, since most things area bit of both.
The MIU approach can be viewed as being based on some specification ofa shopping time model, and the notion of a production function for shoppingtime allows for less rigid substitution between different means of carrying outtransactions. The example in the text emphasized the use of time or moneyfor transactions, but one could allow a variety of means of payment to enterthe production function as imperfect substitutes. Of course that treats the de-gree of substitution as exogenous, which is also unsatisfactory. We would reallylike a model that accounts for why certain means of payment are used in somecircumstances and others in different circumstances.
By emphasizing the link between transactions and money demand, the CIAapproach probably provides the more natural starting point for an analysis ofcredit card usage. For an interesting recent analysis, see D.L. Brito and P.R.Hartley, “Consumer Rationality and Credit Cards,” Journal of Political Economy,103 (2), April 1995, 400-433.
12
5. Consider the model of Section 3.3.1. Suppose that money is required topurchase both consumption and investment goods. The cash-in-advanceconstraint then becomes ct + xt ≤ mt−1/Πt + τ t where x is investment.Assume the aggregate production function takes the form yt = eztkαt n
1−αt .
Show that the steady-state capital-labor ratio is affected by the rate ofinflation. Does a rise in inflation raise or lower the steady-state capital-labor ratio. Explain.
Most of this problem is already worked out as part of the solution to Problem3.b. The model of Section 3.3.1 assumed utility depended only on consumption,so there was no labor-leisure choice. Otherwise, the set-up is similar to Problem3.b, so the equations defining the steady-state are, from (17) - (20),
uc(css) = λss + µss (21)
β
(λss + µss
Πss
)= λss (22)
λss + µss = β [Rssλss + µss(1− δ)] (23)
Equation (19) has been dropped since there is no labor supply decision, andutility in (21) depends only on consumption. From (22),
µss = λss(Πss
β− 1
)
so (23) becomes (Πss
β
)= β
[Rss +
(Πss
β− 1
)(1− δ)
]
For convenience, normalize n to 1. Then Rss + 1 − δ ≡ αyss/kss + 1 − δ =α(kss)α−1 + 1− δ, and equation (23) implies
α(kss)α−1 =
(Πss
β
)[1
β− 1 + δ
]
Hence, the steady-state capital-labor ratio is
kss =
{(1
α
)(Πss
β
)[1
β− 1 + δ
]} −11−α
which is decreasing in the inflation rate (Πss). Higher inflation implies a highertax on capital purchases and this lowers the steady-state stock of capital.
13
6. Consider the following model:
Preferences: Et
∞∑i=0
βi[ln ct+i + b lndt+i]
Budget constraint: ct + dt +mt + kt = Akat−1 + τ t +mt−1
1 + πt+ (1− δ)kt−1
(24)
Cash-in-advance constraint: ct ≤ τ t +mt−1
1 + πt(25)
where m denotes real money balances and πt is the inflation rate fromperiod t − 1 to period t. The two consumption goods, c and d, representcash (c) and credit (d) goods. The net transfer τ is viewed as a lump-sumpayment (or tax) by the household.
(a) Does this model exhibit superneutrality? Explain.
(b) What is the rate of inflation that maximizes steady-state utility?
(a) The model exhibits superneutrality if the real variables k, c, and d areindependent of π in the steady-state. If we define at ≡ τ t +
mt−1
1+πt, the value
function can be defined as
V (at, kt−1) = max {ln ct+i + b lndt+i
+βEtV
(τ t+1 +
mt
1 + πt+1, Akat−1 + (1− δ)kt−1 + at − ct − dt −mt
)}
where the maximization is subject to
ct ≤ at
Let µ denote the Lagrangian multiplier associated with this cash-in-advance con-straint. From the first order conditions for the agent’s decision problem,
1
ct− βEtVk(at+1, kt)− µt = 0 (26)
b
dt− βEtVk(at+1, kt) = 0 (27)
1
1 + πt+1βEtVa(at+1, kt)− βEtVk(at+1, kt) = 0 (28)
Va(at, kt−1) = βEtVk(at+1, kt) + µt (29)
14
Vk(at, kt−1) = β(aAka−1
t−1 + 1− δ)EtVk(at+1, kt) (30)
plus the two constraints (24) and (25). Equation (30) implies that, in the steady-state,
1 = β[(aA(kss)a−1 + 1− δ
)]⇒ kss =
{(1
αA
)[1
β− 1 + δ
]} −11−α
(31)
This means that the steady-state capital stock is independent of the inflationrate.
Let λt ≡ βEtVk(at+1, kt). From (26) and (27),
dtct
=
(1 +
µtλt
)b (32)
Equations (28) and (29) imply
βEt
(λt+1 + µt+1
)1 + πt+1
= λt
In the steady-state, this implies
λss + µss
λss=
(1 +
µss
λss
)=
1 + πss
β
and combining this with (32),
dss
css=
(1 + πss
β
)b (33)
so the relative consumption of c and d depends on the rate of inflation. Thereal equilibrium does not display superneutrality.
(b) From the steady-state marginal product of capital condition (31), we havethe standard result that the real rate of return will equal 1/β. Letting R ≡ 1/β,equation (33) can be written as
dss = (1 + iss)bcss
where i = R(1 + π). Letting Z = A(kss)a − δkss, in the steady-state we havefrom the budget constraint (24) Z = css+ dss = css+(1+ iss)bcss or css = ∆Z
where ∆ = [1 + (1 + iss)b]−1. Hence, steady-state utility of the representativeagent can be expressed as
1
1− β[ln css + b lndss] =
1
1− β[ln css + b ln(1 + iss)bcss]
=1
1− β[ln∆Z + b ln(1 + iss)b∆Z]
=1
1− β[(1 + b) ln∆Z + b ln(1 + iss)b]
=1 + b
1− βln∆+
1 + b
1− βlnZ +
b
1− βln(1 + iss)b
15
Now maximize this with respect to the nominal rate of interest i. Since Z wasshown earlier to be independent of the inflation rate, the first order conditionis
− 1 + b
1− β
(b
1 + (1 + iss)b
)+
1
1− β
[b
1 + iss
]= 0
or
1
1 + iss=
1 + b
1 + (1 + iss)b
which implies
1 + (1 + iss)b
1 + iss= 1 + b
This holds if and only if
iss = 0
So the optimal rate of inflation will be the rate that yields a zero nominal rateof interest.
7. Consider the following model:
Preferences: Et
∞∑i=0
βi (ln ct+i + lndt+i)
Budget constraint: ct + dt +mt + kt = Akat−1 + τ t +mt−1
1 + πt+ (1− δ)kt−1
where m denotes real money balances and πt is the inflation rate fromperiod t− 1 to period t. Utility depends on the consumption of two typesof good; c must be purchased with cash, while d can be purchased usingeither cash or credit. The net transfer τ is viewed as a lump-sum payment(or tax) by the household. If a fraction q of d is purchased using cash,then the household also faces a cash-in-advance constraint that takes theform
ct + qdt ≤ mt−1
1 + πt+ τ t
What is the relationship between the nominal rate of interest and whetherthe cash-in-advance constraint is binding? Explain. Will the householdever use cash to purchase d (i.e. will the optimal q ever be greater thanzero)?
16
The basic model is similar to the one studied in Problem 6, differing only inthe utility function and the cash-in-advance constraint. The value function is
V (at, kt−1) = max {ln ct+i + lndt+i
+βEtV
(τ t+1 +
mt
1 + πt+1, Akat−1 + (1− δ)kt−1 + at − ct − dt −mt
)}
where at = τ t +mt−1
1+πtand ct + qdt ≤ at. and we require that 0 ≤ q ≤ 1
since q is the fraction of the d good purchased with cash. Actually, the relevantconsideration is whether q is positive or not. Let θ denote the Lagrangian onthe constraint q ≥ 0. The first order conditions for the household’s decisionproblem for the current are simply stated here as, modifying them to reflect thedifferent utility function and cash-in-advance constraint:
1
ct− βEtVk(at+1, kt)− µt = 0
1
dt− βEtVk(at+1, kt)− qµt = 0
1
1 + πt+1βEtVa(at+1, kt)− βEtVk(at+1, kt) = 0
Va(at, kt−1) = βEtVk(at+1, kt) + µt
Vk(at, kt−1) = β(aAka−1
t−1 + 1− δ)EtVk(at+1, kt)
In addition, we need the first order condition for the optimal choice of q. Thistakes the form
−µtdt + θt ≤ 0 qtθt = 0
where the condition qtθt = 0 is the complementary slackness condition associatedwith the inequality constraint on q. Since q cannot be reduced below zero, theoptimum can have −µtdt+θt < 0 at q = 0; utility could be increased by reducingq even further, but the non-negativity constraint binds. As long as the nominalrate of interest is positive, µ > 0, and µd > 0. this implies that θ > 0 fromwhich the condition qθ = 0 implies that q = 0. So, as long as the nominal rateof interest is positive, the household will never use cash to purchase d.
8. Trejos and Wright (1993) find that if no search is allowed while bargainingtakes place, output tends to be too low (the marginal utility of outputexceeds the marginal production costs). Show that output is also too lowin a basic cash-in-advance model. (For simplicity, assume only labor isneeded to produce output according to the production function y = n.)Does the same hold true in a money-in-the-utility-function model?
17
In a basic cash-in-advance model, inflation taxes cash goods. Suppose thenominal rate of interest is positive; relative to the case of a zero nominal interestrate, households will be consuming fewer cash goods (which bear the inflationtax) and more credit goods. Since leisure is a credit good, inflation will tend tolower output by increasing the demand for leisure and reducing labor supply. Forexample, equation (3.29) on page 110 shows how inflation reduces labor supplyrelative to the case of a zero nominal rate of interest.
If we modify the model of Section 3.3.2.1 by ignoring capital, and assume theproduction function is y = n, then the value function for the decision problemof the household becomes (see section 3.6 of the Chapter Appendix):
V (at) = max
{u(ct, 1− nt) + βV
(nt + at − ct
Πt+1+ τ t+1
)}
where at = τ t +mt−1/Πt, and the maximization is subjective to the cash-in-advance constraint ct ≤ at. If µt is the Lagrangian multiplier associated withthe cash-in-advance constraint, then the first order necessary conditions are
uc(ct, 1− nt)− βV ′(nt + at − ct −mt)
Πt+1− µt = 0
−ul(ct, 1− nt) +βV ′(nt + at − ct −mt)
Πt+1= 0
V ′(at) =βV ′(nt + at − ct −mt)
Πt+1+ µt
Let λt ≡ βV ′(nt+at−ct−mt)Πt+1
. Then these first order conditions imply
ul(ct, 1− nt)
uc(ct, 1− nt)=
λtλt + µt
=
(1 +
µtλt
)−1
≤ 1
As long as the cash-in-advance constraint is binding, µ > 0 and ul/uc is greaterthan it would be in the case in which µ = 0. Since ul/uc is increasing in laborsupply, labor supply and output is reduced relative to the µ = 0 case. In thisframework, the marginal cost of output is ul since this is the utility cost ofsupplying additional labor. The marginal utility of the output that is producedis uc. Since ul < uc when the cash-in-advance constraint binds, the marginalutility of output exceeds the marginal cost of production.
In a basic money-in-the-utility-function model, the relevant condition wasgiven by equation (2.34) on page 66. The marginal utility cost of supplyingmore labor ul is just equal to the marginal utility of consumption times thatadditional output produced fnuc. So the marginal cost of production and themarginal utility of output are equal. This doesn’t mean money and inflationdon’t affect output. A positive nominal interest rate reduces real money holdings
18
relative to the social optimum. How that affects labor supply (and output) willdepend on how a decrease in m affects ul/uc and the effect could go either way.For the utility function used in the linear version of the money-in-the-utility-function model of Chapter 2, equation (2.45) shows that a lower value of m will,for given c and y, act to increases labor supply for Φ < 1 and decrease laborsupply for Φ > 1. Thus, if, for example, Φ < 1, consumption and money arecomplements; an increase in m increases the marginal utility of consumption.Higher inflation that reduces m also leads to a fall in the marginal utility ofconsumption. Households will shift towards consuming more leisure and fewerconsumption goods. The decline in labor supply as more leisure is consumed willlower output.
19
3 Chapter 4: Money and Public Finance
1. Consider the version of the Sidrauski (1967) model studied in Problem1 of Chapter 2. Utility was given by u(ct,mt) = w(ct) + v(mt), withw(ct) = ln ct and v(mt) = mt (B −D lnmt) where B and D are positiveparameters. Steady-state revenue from seigniorage is given by θm, whereθ is the growth rate of the money supply.
(a) Is there a “Laffer Curve” for seigniorage (i.e. are revenues increasingin θ for all θ ≤ θ∗ and decreasing in θ for all θ > θ∗ for some θ∗?
(b) What rate of money growth maximizes steady-state revenues fromseigniorage?
(c) Assume now that the economy’s rate of population growth is n andreinterpret m as real money balances per capita. What rate of infla-tion maximizes seigniorage? How does it depend on n?
(a) From Problem Set 2, we know that the demand for money in this modelis given by mt = Ae−i/ctD where lnA = B
D − 1. Hence, seigniorage in thesteady-state is equal to
sss = πAe−(rss+π)/cssD
Taking the derivative with respect to π,
∂sss
∂π= Ae−(rss+θ)/cssD − π
cssDAe−(rss+π)/cssD = Ae−(rss+π)/cssD
(1− π
cssD
)
This is positive (i.e. seigniorage is increasing in inflation) for π < cssD, andnegative for π > cssD. Hence, there is a Laffer curve.
(b) Steady-state seigniorage is maximized for π = θ = cssD.
(c) With population growth at the rate γ, the growth rate of per capital moneybalances is given by
θ − π − γ
Hence, in the steady-state, π = θ − γ. Steady-state seigniorage will be still bemaximized at an inflation rate of cssD, but this now corresponds to a rate ofmoney growth of cssD+ γ.
2. Suppose that government faces the following budget identity:
bt = Rbt−1 + gt − τ tyt − st
where the terms are one period debt, gross interest payments, govern-ment purchases, income tax receipts and seigniorage. Assume seigniorage
20
is given by f(πt) where π is the rate of inflation. The interest factorR is constant and the expenditure process {gt+i}∞i=0 is exogenous. Thegovernment sets time paths for the income tax rate and for inflation tominimize
Et
∞∑i=0
βi [h(τ t+i) + k(πt+i)]
where the functions h and k represent the distortionary costs of the twotax sources. Assume the functions h and k imply positive and increasingmarginal costs of both revenue sources.
(a) What is the intratemporal optimality condition linking the choices ofτ and π at each point in time?
(b) What is the intratemporal optimality condition linking the choice πat different points in time?
(c) Suppose y = 1, f(π) = aπ, h(τ) = bτ2 and k(π) = cπ2. Evaluate theinter- and intratemporal conditions. Find the optimal settings for τ tand πt in terms of bt−1 and
∑R−igt+i.
(d) Using your results from part (c), when will optimal financing implyconstant planned tax rates and inflation over time?
(a) Solving the budget constraint forward, the government’s decision problemcan be written as minEt
∑∞i=0 β
i (h(τ t+i) + k(πt+i)) subject to
Rbt−1 +Et
∑R−igt+i −Et
∑R−i [τ t+iyt+i + f(πt+i)] = 0
Let λ be the Lagrangian multiplier associated with this constraint. The firstorder conditions are
Et
[βih′(τ t+i)− λR−iyt+i
]= 0
and
Et
[βik′(πt+i)− λR−if ′(πt+i)
]= 0
Hence, the condition linking taxes and seigniorage at each date t + i take theform
Eth′(τ t+i)
Etyt+i=
λ
(βR)i=Etk
′(πt+i)
Etf ′(πt+i)
(b) The first order conditions for seigniorage at dates t + i and t + j takethe form βiEtk
′(st+i) = λR−iEtf′(πt+i) and βjEtk
′(st+j) = λR−jEtf′(πt+j),
or
Et(βR)i k
′(πt+i)
f ′(πt+i)= λ = Et(βR)
j k′(πt+j)
f ′(πt+j)
21
(c) Given the assumed functional forms, the first order conditions become
2bEtτ t+i =λ
(βR)i= 2( ca
)Etπt+i
which implies that Etτ t+i =(
cab
)Etπt+i for all i. The intertemporal condition
becomes
2( ca
)Etπt+i(βR)
i = 2( ca
)Etπt+j(βR)
j or for j = 0, Etπt+i = (βR)−iπt
These results imply Etτ t+i =cab(βR)
−iπt.Now we can evaluate the government’s budget constraint recalling at y = 1):
Rbt−1 +Et
∑R−igt+i = Et
∑R−i(τ t+iyt+i + f(πt+i))
=∑
R−i(c
ab(βR)−iπt + a(βR)−iπt)
=( cab
+ a)πt∑
(βR2)−i
This implies that
πt =( cab
+ a)−1
B[Rbt−1 +Et
∑R−igt+i
]where B = βR2/(βR2 − 1).
Finally, the optimal tax rate is given by
τ t =( cab
)πt =
( cab
)( cab
+ a)−1
B[Rbt−1 +Et
∑R−igt+i
](d) Etπt+i = (βR)−iπt = πt if and only if βR = 1 (i.e. R = 1/β).
3. Mankiw (1987) suggested that the nominal interest rate should evolve as arandom walk under an optimal tax policy. Suppose the real rate of interestis constant and that the equilibrium price level is given by equation (4.24).Suppose the nominal money supply is given by mt = mp
t + vt where mpt is
the central bank’s planned money supply and vt is a white noise controlerror. Let θ be the optimal rate of inflation. There are different processesformp that lead to the same average inflation rate but different time seriesbehavior of the nominal interest rate. For each of the processes for mp
t
given in a and b, show that average inflation is θ; also show whether thenominal interest rate is a random walk.
(a) mpt = θ(1− γ)t+ γmt−1;
(b) mpt = mt−1 + θ.
22
Equation (4.24) states that
pt =mt
1 + α+αEtpt+1
1 + α(34)
For the money process in part (a), this becomes
pt =θ(1− γ)t+ γmt−1 + vt
1 + α+αEtpt+1
1 + α(35)
and the no-bubbles solution is of the form
pt = p0 + at+ bmt−1 + cvt
where a, b, and c are coefficients to be determined. This solution implies
Etpt+1 = p0 + a(t+ 1) + bmt = p0 + a(t+ 1) + b [θ(1− γ)t+ γmt−1 + vt]
Using this and the trial solution in equation (35) yields
p0 + at+ bmt−1 + cvt =θ(1− γ)t+ γmt−1 + vt
1 + α
+α [p0 + a(t+ 1) + b (θ(1− γ)t+ γmt−1 + vt)]
1 + α
=
(θ(1− γ) + αa+ αbθ(1− γ)
1 + α
)t+
α(p0 + a)
1 + α
+
(γ(1 + αb)
1 + α
)mt−1 +
(1 + αb
1 + α
)vt
This will hold for all realizations of vt and mt−1 if
p0 =α(p0 + a)
1 + α⇒ p0 = αa
a =
(θ(1− γ) + αa+ αbθ(1− γ)
1 + α
)⇒ a = θ(1− γ)(1 + αb)
b =γ(1 + αb)
1 + α⇒ b =
γ
1 + α(1− γ)
c =1
1 + α(1− γ)
Substituting for b in the expression for a,
a = θ(1− γ)
(1 +
αγ
1 + α(1− γ)
)= θ(1− γ)
(1 + α
1 + α(1− γ)
)
23
Hence, the equilibrium price level evolves according to
pt = θ(1− γ)
(1 + α
1 + α(1− γ)
)(α+ t) +
(1
1 + α(1− γ)
)(γmt−1 + vt)
Average inflation will equal
∆pt = θ(1− γ)
(1 + α
1 + α(1− γ)
)+
γ∆mt−1
1 + α(1− γ)
= θ(1− γ)
(1 + α
1 + α(1− γ)
)+
γθ
1 + α(1− γ)
= θ
Expected inflation is equal to
Etpt+1 − pt = θ(1− γ)
(1 + α
1 + α(1− γ)
)+
(1
1 + α(1− γ)
)(γmt − γmt−1 − vt)
= θ(1− γ)
(1 + α
1 + α(1− γ)
)+
(1
1 + α(1− γ)
)(γ∆mt − vt)
With a constant real rate of interest, as was assumed in deriving equation (4.24),the nominal rate of interest will equal
it = r0 + θ(1− γ)
(1 + α
1 + α(1− γ)
)+
(1
1 + α(1− γ)
)(γ∆mt − vt)
= i0 +
(1
1 + α(1− γ)
)(γ∆mt − vt)
Since ∆mt = θ(1−γ)+γ∆mt−1+vt, ∆mt is a first order autoregressive processand (1− γL)∆mt = θ(1− γ) + vt. So the nominal interest rate is of the form
it = i0 + zt − vt
where zt AR(1) and vt is white noise. Quasi-first differencing,
(1− γL)it = (1− γ)i0 + (1− γL)(zt − vt)
= (1− γ)i0 + θ(1− γ) + vt − (1− γL)vt
= (1− γ)i0 + θ(1− γ) + γvt−1
Thus, the nominal interest rate, so the nominal interest rate will follow the firstorder autoregressive process
it = i′0 + γit−1 + γvt−1
With the money supply process in (b), equation (4.24) becomes
pt =θ +mt−1 + vt
1 + α+αEtpt+1
1 + α(36)
24
and the equilibrium solution for pt is of the form pt = p0 + bmt−1 + cvt. Usingthis in (36),
p0 + bmt−1 + cvt =θ +mt−1 + vt
1 + α+α [p0 + b(θ +mt−1)]
1 + α
=
(1 + αb
1 + α
)θ +
αp01 + α
+
(1 + αb
1 + α
)mt−1 +
(1
1 + α
)vt
or p0 = (1 + α)θ, b = 1, and c = 1/(1 + α). With
pt = (1 + α)θ +mt−1 +1
1 + αvt
average inflation is just θ. Expected inflation is
Etpt+1 − pt =mt −mt−1 +1
1 + αvt = θ +
(1
1 + α
)vt
With a constant real rate of interest, as was assumed in deriving equation(4.24), the nominal rate of interest will equal
it = r0 + θ +
(1
1 + α
)vt
so that it is equal to a constant plus a white noise error; it is not a randomwalk.
4. Suppose the Correia-Teles model of Section 4.5.3 is modified so that outputis equal to f(n) where f is a standard neoclassical production functionexhibiting positive but diminishing marginal productivity of n. If f(n) =na for 0 < a < 1, does the optimality condition given by (4.44) continuesto hold?
Start with the budget constraint when f(n) = na. Equation (4.42) becomes
dt−1 =∞∑i=0
Di
[ct+i − (1− τ t+i)f(1− lt+i − nst+i) +RtIt+imt−1+i
]
=∞∑i=0
Di
[ct+i − (1− τ t+i)(1− lt+i − nst+i)
a +RtIt+imt−1+i
]
while the first order condition for labor supply, equation (4.39), is modified tobecome
ul(1− τ t)
= aλt(1− lt − nst )a−1 (37)
25
Following the same steps outlines on page 170, we use the fact that dt−1 = 0and Di = βiD0λt+i/λt to obtain
D0
λt
∞∑i=0
βi[λt+ict+i − λt+i(1− τ t+i)(1− lt+i − nst+i)
a − λt+iIt+1+imt+i
]= 0
From (37), λt+i(1−τ t+i) = ul(1−lt−nst )1−a/a. Making this substitution (alongwith the others discussed in the text) yields
0 =D0
λt
∞∑i=0
βi[ucct+i − ulηgn
st+i −
(ul(1− lt − nst )
1−a
a
)(1− lt+i − nst+i)
a
]
=D0
λt
∞∑i=0
βi[ucct+i − ulηgn
st+i −
(ula
)(1− lt − nst )
]
This implies
∞∑i=0
βi[ucct+i −
(ula
)(1− lt)− ul
(1
a− η
)nst+i
]= 0
which corresponds to equation (4.43) on page 171. Notice that the only modifi-cation is that ns is multiplied by the factor 1
a − η; in the example of the text,a = 1 and this became 1− η.
The first-order condition for the optimal choice of m in the social welfareproblem is [
βiψul(1
a− η)− µt+i
]g′ = 0
which replaces (4.44). As long as βiψul(1a − η) − µt+i must be nonzero, the
optimum still involves g′ = 0, or a zero nominal interest rate.
26
4 Chapter 5: Money and Output in the Short
Run
1. Assume household preferences are given by U =
[c(M
P )b]1−Φ
1−Φ +Ψ (1−N)1−η
1−η
and aggregate output is given by Y = KαN1−α. Linearize around thesteady-state the labor market equilibrium condition equation (5.26) fromthe monopolistic competition model. How does the result depend on q?Explain.
Equation (5.26) states that
Ul
Uc= qMPL
For the functional forms specified in the question, this becomes
(1−N)−η
c−Φ(mb)1−Φ= q(1− α)
Y
N(38)
where m =M/P .Using the methods employed in the text, we can define Xss as the steady-
state value of X and x as the percent deviation of x around its steady-statevalue, so x = xss(1 + x). Then, (38) can be written as[
Lss(1 + l)]−η
[css(1 + c)]−Φ[(mss(1 + m))b
]1−Φ= q(1− α)
Y ss(1 + y)
Nss(1 + n)(39)
where L = 1−N . Since
(Lss)−η
(css)−Φ[(mss)b
]1−Φ= q(1− α)
Y ss
Nss
equation (39) becomes
(1 + l)−η
(1 + c)−Φ [(1 + m)b]1−Φ
=(1 + y)
(1 + n)(40)
Notice that q has dropped out; the labor market deviations around the steady-state will not be affected by q. Now take logs of (40) and employ the approxi-mation that ln(1 + x) ≈ x for small x, to obtain
−ηl +Φc− b(1−Φ)m = y − n
Since Lss(1+ l) = 1−Nss(1+ n), this implies l = −Nss
Lss n. Using this we have,
−η(−N
ss
Lssn
)+Φc− b(1−Φ)m = y − n
27
or
n =
(1
1 + ηNss
Lss
)(y −Φc+ b(1−Φ)m)
2. The Chari, Kehoe, and McGratten (1996) model of price adjustment ledto equation (5.30). Using equation (5.29), show that the parameter a in(5.30) equals (1−√
γ)/(1 +√γ).
Equation (5.29), page 200, states that
pt =1
2
(1− γ
1 + γ
)[pt−1 +Etpt+1] +
(γ
1 + γ
)[mt +Etmt+1] (41)
If m follows a random walk as was assumed in deriving (5.30), Etmt+1 = mt
and (5.29) becomes
pt =1
2
(1− γ
1 + γ
)[pt−1 +Etpt+1] +
(2γ
1 + γ
)mt
It will be convenient to define b ≡ 12
(1−γ1+γ
)and rewrite this as
pt = b [pt−1 +Etpt+1] + (1− 2b)mt (42)
Let the proposed solution be
pt = a1pt−1 + a2mt
From (5.30), Etpt+1 = a1pt + a2mt = a1 (a1pt−1 + a2mt) + a2mt, so equation(42) becomes
pt = b [pt−1 + a1 (a1pt−1 + a2mt) + a2mt] + (1− 2b)mt
= b(1 + a21
)pt−1 + [1− 2b+ ba2(1 + a1)]mt
For this to equal the proposed solution for all realizations of pt−1 and mt requiresthat
a1 = b(1 + a21
)and
a2 = 1− 2b+ ba2(1 + a1) ⇒ a2 =1− 2b
1− b(1 + a1)
The first of these conditions requires that a1 be the solution to
a21 −(1
b
)a1 + 1 = 0
28
or
a1 =b−1 ±√
b−2 − 4
2
Recalling that b was equal to 12
(1−γ1+γ
), this becomes
a1 =2(
1+γ1−γ
)±√4(
1+γ1−γ
)2− 4
2
=
(1 + γ
1− γ
)±√(
1 + γ
1− γ
)2
− 1
=
(1 + γ
1− γ
)±(
1
1− γ
)√1 + 2γ + γ2 − (1− 2γ + γ2)
=
(1 + γ
1− γ
)± 2
(1
1− γ
)√γ =
(1−√
γ)2
1− γand
(1 +
√γ)2
1− γ
Since 1− γ = (1 +√γ)(1−√
γ), we can write these two potential solutions as(1−√
γ)2
1− γ=
1−√γ
1 +√γ
and (1 +
√γ)2
1− γ=
1 +√γ
1−√γ
Both only the first of these is less than 1 in absolute value, so the stable solutionhas
a1 =1−√
γ
1 +√γ
Returning to the condition for a2, and using the value for a1 just found,together with the definition of b,
a2 =1− 2b
1− b(1 + a1)=
1−(1−γ1+γ
)1− 1
2
(1−γ1+γ
)(1 +
1−√γ
1+√γ
)
=
2γ1+γ
1−(1−γ1+γ
)(1
1+√γ
) =2γ
1 + γ −(
1−γ1+
√γ
)=
2γ
γ +√γ= 1−
(1−√
γ
1 +√γ
)
which verifies that a2 = 1− a1.
29
3. Equation (5.28) was obtained from equation (5.27) by assuming R = 1.Show that in general,
pt =
(Rss
1 +Rss
)[pt +
1
RssEtpt+1
]+
(Rss
1 +Rss
)[vt +
1
RssEtvt+1
]
Equation (5.27) of the text states that
Pt =Et
[P θt VtYt +R−1
t+1Pθt+1Vt+1Yt+1
]qEt
[P
11−q
t Yt +R−1t+1P
11−q
t+1 Yt+1
] (43)
If we evaluate this at the steady state, recalling that θ = (2− q)/(1− q),
P ss =(P ss)θ V ssY ss
[1 + (Rss)−1
]q (P ss)
11−q Y ss
[1 + (Rss)−1
] =1
qP ssV ss (44)
Let lower case letters denote percentage deviations from the steady-state, so1+x = Xt/X
ss. Then the left side of equation (5.27) can be written P ss(1+ pt)while the right side becomes
(P ss)θ V ssY ss[(1 + pt)θ(1 + vt)(1 + yt) + (Rss)−1Et(1 + rt+1)−1(1 + pt+1)θ(1 + vt+1)(1 + yt+1)
](P ss)
11−q qY ss
[(1 + pt)
11−q (1 + yt) + (Rss)−1Et(1 + rt+1)−1(1 + pt+1)
11−q (1 + yt+1)
]
Now using the approximations (1+x)s ≈ 1+ sx and (1+x)(1+ z) ≈ 1+x+ z,this becomes
P ssV ss[(1 + θpt + vt + yt) + (Rss)−1Et(1 + θpt+1 + vt+1 + yt+1 − rt+1)
]q[(
1 +(
11−q
)pt + yt
)+ (Rss)
−1Et
(1 +(
11−q
)pt+1 + yt+1 − rt+1
)]
After some cancellation, equation (43) can be written
(1 + pt) =
[(1 + θpt + vt + yt) + (Rss)−1 Et(1 + θpt+1 + vt+1 + yt+1 − rt+1)
][(
1 +(
11−q
)pt + yt
)+ (Rss)−1Et
(1 +(
11−q
)pt+1 + yt+1 − rt+1
)](45)
This expression is of the form
1 + x =1 + z +R−1(1 + d)
1 + s+R−1(1 + c)(46)
30
which can be written as
(1 + x)[1 + s+R−1(1 + c)
]= 1 + z +R−1(1 + d)
Multiplying out the left side,
1 + x+ s+ sx+R−1(1 + x) +R−1(x+ xc) ≈ 1 + x+ s+R−1(1 + c) +R−1x
=(1 +R−1
)x+ 1 + s+R−1 (1 + c)
yielding for equation (46),(1 +R−1
)x+ 1 + s+R−1 (1 + c) ≈ 1 + z +R−1(1 + d)
or
x ≈ z +R−1(1 + d)− s−R−1 (1 + c)
(1 +R−1)=
(R
1 +R
)[z − s+
(1
R
)(d− c)
]
Returning to equation (45), we have x = pt, z = θpt+vt+yt, s =(
11−q
)pt+yt,
d = Et (θpt+1 + vt+1 + yt+1 − rt+1), and d = Et
[(1
1−q
)pt+1 + yt+1 − rt+1
], so
pt =
(Rss
1 +Rss
)[pt + vt +
(1
Rss
)Et (pt+1 + vt+1)
]
since θ −(
11−q
)= 1. This is our desired result.
4. Using the equilibrium condition (5.42) for the price level, show that equi-librium output is independent of any policy response to εt−1 or vt−1.
Equation (5.42) states that
pt =d2mt + a(1 + d2)Et−1pt +Etpt+1 − d2vt + ut − (1 + d2)εt
(1 + a)(1 + d2)(47)
Suppose that monetary policy does respond to εt−1 and vt−1 by setting the nom-inal supply of money according to
mt = b1εt−1 + b2vt−1
Substituting this expressing into (47),
pt =d2 (b1εt−1 + b2vt−1) + a(1 + d2)Et−1pt +Etpt+1 − d2vt + ut − (1 + d2)εt
(1 + a)(1 + d2)(48)
31
Assuming all the disturbance terms are serially uncorrelated, we can use themethod of undetermined coefficients to find the solution for the equilibrium pricelevel. Inspection of (48) suggests the following guess:
pt = γ0 + γ1εt−1 + γ2vt−1 + γ3vt + γ4ut + γ5εt (49)
where xt ≡ −d2vt + ut − (1 + d2)εt. Using (49),
Et−1pt = γ0 + γ1εt−1 + γ2vt−1 (50)
and
Etpt+1 = γ0 + γ1εt + γ2vt (51)
Substituting equations (49) - (51) into (48), yields, with some rearranging,
(1 + a)(1 + d2)pt = d2 (b1εt−1 + b2vt−1) + a(1 + d2) (γ0 + γ1εt−1 + γ2vt−1)
+γ0 + γ1εt + γ2vt − d2vt + ut − (1 + d2)εt
= γ0 [1 + a(1 + d2)] + [d2b1 + aγ1(1 + d2)] εt−1
+ [d2b2 + γ2a(1 + d2)] vt−1
+ [γ1 − (1 + d2)] εt + [γ2 − d2] vt + ut (52)
We could now replace pt on the left side of this equation with the proposedsolution given in equation (49) and equate coefficients to solve for the values ofthe γi coefficients. However, to determine the effect of the policy reaction onoutput, we do not need to do this. From equation (5.34) on page 205, outputdepends on the price surprise term pt − Et−1pt. We can obtain this by takingexpectations of (52) based on t−1 information and subtract the result from (52).So first taking expectations,
(1 + a)(1 + d2)Et−1pt = γ0 [1 + a(1 + d2)]
+ [d2b1 + aγ1(1 + d2)] εt−1 + [d2b2 + γ2a(1 + d2)] vt−1
Subtracting this from (52),
(1 + a)(1 + d2) (pt − Et−1pt) = [γ1 − (1 + d2)] εt + [γ2 − d2] vt + ut
This is independent of the policy response coefficients b1 and b2.Because any systematic policy response to εt−1 or vt−1 is fully incorporated
into the public’s expectations at the start of period t, it cannot generate any pricesurprise; pt (and Et−1pt) adjust fully to anticipated or predictable movementsin the period t nominal supply of money.
5. Assume nominal wages are set for one period but that they can be indexedto the price level:
wct = w0
t + b(pt − Et−1pt)
where w0 is a base wage and b is the indexation parameter (0 ≤ b ≤ 1).
32
(a) How does this change modify the aggregate supply equation given by(5.18)?
(b) Assume the indexation parameter is set to minimize Et−1(nt − n)2.Using your modified aggregate supply equation, together with (5.35)- (5.37) and a money supply process mt = ωt, show that the opti-mal degree of wage indexation is increasing in the variance of ω anddecreasing in the variance of ε (Gray 1978).
(a) From equation (5.16) and the new specification for the contract wage,employment is given by
nt = yt −[w0t + b(pt − Et−1pt)
]+ pt
If the base wage is set according to (5.15), w0t = Et−1 (yt + pt − nt), and
nt − Et−1nt = yt − Et−1yt + (1− b) (pt − Et−1pt) (53)
Notice that if b = 0 (no indexation), we obtain the expression in the test (equa-tion 5.16). At the other extreme, if b = 1, nominal wages are completely indexedand adjust fully to unexpected changes in the price level. As a result, the realwage and employment are insulated from price level movements.
Since the model underlying equations (5.34) - (5.37) was based on the as-sumption that labor supply was inelastic, we can set Et−1nt = 0 since all vari-ables should be interpreted as deviations around a steady-state.
Substituting (53) into the production function (5.8),
yt − Et−1yt = (1− α) [yt − Et−1yt + (1− b) (pt − Et−1pt)] + zt − Et−1zt
= a(1− b) (pt − Et−1pt) + εt (54)
where, as in the text, a = (1− α)/α and εt = (zt − Et−1zt) /α. Equation (54)shows that, relative to (5.18), the effect of a price surprise on output is nowa(1− b) < a.
(b) To solve for the variance of employment, use (5.35) - (5.37) to solve foryt and pt, using the assumption that mt = ωt. From (5.36) and (5.37),
rt = it + pt − Et−1pt
= d2 (yt + pt + vt − ωt) + pt − Et−1pt
Substituting this into the aggregate spending equation (5.35),
yt = Etyt+1 − [d2 (yt + pt + vt − ωt) + pt − Et−1pt] + ut (55)
=Etyt+1 − [(1 + d2)pt − Et−1pt] + ut + d2(ωt − vt)
1 + d2
If we assume the productivity shock z is serially uncorrelated, then Etyt+1 =Etzt+1 = 0 and (55) implies
yt − Et−1yt = − (pt − Et−1pt) + st (56)
33
where st ≡ [ut + d2(ωt − vt)] /(1 + d2). Solving (54) and (56) for yt − Et−1ytand pt − Et−1pt,
yt − Et−1y =
(a(1− b)
1 + a(1− b)
)st +
1
1 + a(1− b)εt
pt − Et−1pt =
(st − εt
1 + a(1− b)
)
Using these results in (53),
nt − Et−1nt = yt − Et−1yt + (1− b) (pt − Et−1pt)
=
((1− b)(1 + a)
1 + a(1− b)
)st +
(b
1 + a(1− b)
)εt (57)
so
Et−1(nt − n)2 = Et−1
[((1− b)(1 + a)
1 + a(1− b)
)st +
(b
1 + a(1− b)
)εt − n
]2
=
((1− b)(1 + a)
1 + a(1− b)
)2
σ2s +
(b
1 + a(1− b)
)2
σ2ε + n2
The value of the indexation parameter is picked to minimize this expression.The first order condition for this problem is
2
(−(1− b)(1 + a)2
[1 + a(1− b)]3
)σ2s + 2
((1 + a)b
[1 + a(1− b)]3
)σ2ε = 0
which implies
−(1− b)(1 + a)σ2s + bσ2
ε = 0
or the optimal degree of indexation is
b∗ =(1 + a)σ2
s
(1 + a)σ2s + σ2
ε
= 1− σ2ε
(1 + a)σ2s + σ2
ε
Hence,
0 ≤ b∗ ≤ 1
with the inequalities strict if both σ2ε and σ2
s are positive. Define γ ≡ σ2ε/σ
2s as
the relative variance of productivity shocks to demand side shocks (arising frommoney supply shocks ω, money demand shocks v, and aggregate spending shocksu). Then
b∗ = 1− γ
1 + a+ γ
34
which is decreasing in γ. As aggregate demand shocks become more important(and γ falls), it is optimal to have a higher degree of nominal wage indexation toinsulate real wages and employment from fluctuating. Real productivity shocksdo call for real wage adjustments, so if ε shocks are important, then the optimaldegree of indexation is lower in order to allow for some real wage movementsas the price level changes.
6. The basic Taylor model of price level adjustment was derived under theassumption that the nominal wage set in period-t remained unchanged forperiods t and t + 1. Suppose instead each period t contract specifies anominal wage x1t for period t and x2t for period t + 1. Assume these aregiven by x1t = pt + κyt and x2t = Etpt+1 + κEtyt+1. The aggregate pricelevel at time t is equal to pt =
12(x
1t +x2t−1). If aggregate demand is given
by yt = mt − pt and mt = m0 + ωt, what is the effect of a money shockωt on pt and yt? Explain why output shows no persistence after a moneyshock.
From the definition of the aggregate price level and the contract nominalwages,
pt =1
2[pt + κyt +Et−1pt + κEt−1yt]
= κyt +Et−1pt + κEt−1yt (58)
which can be compared to the equation at the bottom of page 216. Notice thatpt−1 does not appear in (58), since x2t−1 is now set based on Et−1pt rather thanon pt−1 and Et−1pt as in the specification given by equation (5.44).
Substituting the assumed specification for aggregate demand into (58),
pt = κ (mt − pt) + Et−1pt + κEt−1 (mt − pt)
=κmt + (1− κ)Et−1pt + κm0
1 + κ(59)
where use has been made of the fact that Et−1mt = m0 under the assumedmoney supply process. We can write the solution to this as
pt = γ0 + γ1ωt
for γ0 and γ1 such that
γ0 + γ1ωt =κ (m0 + ωt) + (1− κ)γ0 + κm0
1 + κ
or
γ1 =κ
1 + κ
35
and
γ0 = m0
Aggregate output is then given by
yt = mt − pt = m0 + (1− γ1)ωt − γ0 =
(1
1 + κ
)ωt
Since output is equal to the white noise error ω, it displays no persistence.In the standard formulation, some nominal wages in effect during period t
were set in earlier periods on the basis of the price level in those earlier periods.This imparts sluggishness to the adjustment of the price level; pt can no longerjump to fully offset any change in the period t nominal supply of money. Withthe alternative specification used in this problem, nominal wages in effect inperiod t depend only on period t prices and previous expectations about pt. Thus,output in period t is only affected by movements in mt that were unpredictablewhen the oldest contract still in effect was set.
7. The p-bar model led to the following two equations for pt and yt:
pt =d2mt +Etpt+1 − d2vt + ut
1 + d2− yt
yt =d2 (mt − Et−1mt) + Etpt+1 − Et−1pt+1 − d2vt + ut
1 + d2+ (1− γ)yt−1
Assume γ = 1 and
mt =mt−1 + aut
Show that the variance of yt depends on the parameter a. What value ofa would minimize the impact of IS shocks (u) on output?
If γ = 1, lagged output drops out of the model, and from the assumed processfor money, mt−Et−1mt = mt−1+aut−mt−1 = aut. Thus, the output equationcan be written as
yt =Etpt+1 − Et−1pt+1 − d2vt + (1 + ad2)ut
1 + d2(60)
Notice that as long as neither Etpt+1 nor Et−1pt+1 are affected by an IS shock,the impact of u on output would be neutralized if a were set equal to −1/d2. Tocheck whether price expectations are affected, use the price and output equations,together with the money supply process to obtain
(1 + d2)pt = d2 (mt−1 + aut) + Etpt+1 − d2vt + ut
− [Etpt+1 − Et−1pt+1 − d2vt + (1 + ad2)ut] (61)
36
or
(1 + d2)pt = d2 (mt−1 + aut) + Et−1pt+1 − ad2ut (62)
Consider the following proposed solution for pt:
pt = δ0 + δ1mt−1 + δ2ut
Based on this solution,
pt+1 = δ0 + δ1mt + δ2ut+1
= δ0 + δ1 (mt−1 + aut) + δ2ut+1
So
Etpt+1 = δ0 + δ1 (mt−1 + aut)
and
Et−1pt+1 = δ0 + δ1mt−1
Substituting these expressions into (62),
(1 + d2)pt = d2 (mt−1 + aut) + Et−1pt+1 − ad2ut
= d2 (mt−1 + aut) + δ0 + δ1mt−1 − ad2ut
= δ0 + (δ1 + d2)mt−1
Using the proposed solution to eliminate pt,
(1 + d2) [δ0 + δ1mt−1 + δ2ut] = δ0 + (δ1 + d2)mt−1
equating coefficients implies
(1 + d2)δ0 = δ0 ⇒ δ0 = 0
(1 + d2)δ1 = (δ1 + d2) ⇒ δ1 = 1
δ2 = 0
so pt = mt−1.We can now collect these results to evaluate equation (60) for output:
yt =Etpt+1 − Et−1pt+1 − d2vt + (1 + ad2)ut
1 + d2=
[1 + a(1 + d2)]ut − d2vt1 + d2
and the variance of output is
σ2y =
(1 + a(1 + d2)
1 + d2
)2
σ2u +
(d2
1 + d2
)2
σ2v
37
Thus, to insulate output from demand shock, a should be set equal to
a∗ = − 1
1 + d2< 0
Notice that this is a smaller response than found earlier when the possible effectsof u on expected future prices were ignored (see equation 60). A positive ut thatresults in a fall in mt causes private agents to revise downward their forecastof the future price level: Etpt+1 − Et−1pt+1 = aut if a is negative. But from(60), this acts to reduce yt. Thus, to stabilize yt, the reduction in mt needs tobe smaller than would be the case if price expectations did not matter.
8. Derive the equilibrium expression for pt and yt corresponding to equa-tions (5.39) and (5.41) for the case in which the aggregate productivitydisturbance is given by zt = ρzt−1 + et, −1 < ρ < 1.
The equations of the model that led to equations (5.39) and (5.40) were givenby (5.34) - (5.37) and are repeated here:
Aggregate supply: yt = Et−1yt + a(pt − Et−1pt) + εt (63)
Aggregate demand: yt = Etyt+1 − rt + ut (64)
Money demand: mt − pt = yt − d−12 it + vt (65)
Fisher equation: it = rt +Etpt+1 − pt = rt +Etπt+1 (66)
As discussed in the text (page 208), equations (64) - (66) can be combined toyield equation (5.38) of the text:
yt =d2(mt − pt) + Etπt+1 +Etyt+1 − d2vt + ut
1 + d2(67)
As discussed on page 204 (and in footnote 43),
Etyt+1 = Etzt+1
Using the assumed process for z specified in the question,
Etyt+1 = ρzt
Equation (67) then becomes
yt =d2(mt − pt) + Etπt+1 + ρzt − d2vt + ut
1 + d2
38
Equating this expression for yt with the expression for yt given by the aggregatesupply relationship (63) and solving for pt (and using the fact that ρzt−1+et =zt) results in
pt =d2mt + a(1 + d2)Et−1pt +Etπt+1 + ρzt − d2vt + ut − (1 + d2) (ρzt−1 + εt)
d2 + a(1 + d2)
which corresponds to (5.39).Taking expectations of this expression as of time t − 1 and subtracting the
result from pt yields
pt − Et−1pt =d2 (mt − Et−1mt) + Etπt+1 − Et−1πt+1 + ρet − d2vt + ut − (1 + d2)εt
d2 + a(1 + d2)
so that real output, from (63) is
yt = Et−1yt + εt
+a
(d2 (mt − Et−1mt) + Etπt+1 − Et−1πt+1 + ρet − d2vt + ut − (1 + d2)εt
d2 + a(1 + d2)
)
= ρzt−1 +
(d2
d2 + a(1 + d2)
)εt
+a [d2 (mt − Et−1mt) + Etπt+1 − Et−1πt+1 + ρet − d2vt + ut]
d2 + a(1 + d2)
where the only difference from equation (5.41) is the presence of ρet. This is theinnovation in zt that persists into period t + 1. This affects aggregate demandin period t under the assumption that agents are forward looking.
9. Suppose that the nominal money supply evolves according to mt = µ +γmt−1 + ωt for 0 < γ < 1 and ωt a white noise control error. If the restof the economy is characterized by equations (5.34) - (5.37), solve for theequilibrium expressions for the price level, output, and the nominal rateof interest. What is the effect of a positive money shock (ωt > 0) on thenominal rate? How does this result compare to the γ = 1 case discussedin the text? Explain.
To answer this problem, make use of the results in Section 5.7.3 of the Ap-pendix to Chapter 5. Solving the basic model leads to equation (5.85) for theprice level:
pt =d2mt + a(1 + d2)Et−1pt +Etpt+1 − d2vt + ut − (1 + d2)εt
(1 + a)(1 + d2)(68)
39
Given the assumed process for mt, guess that the equilibrium price level is givenby
pt = b0 + b1mt−1 + b2ut + b3εt + b4vt + b5ωt
Based on this guess,
Et−1pt = b0 + b1mt−1
and
Etpt+1 = b0 + b1 (µ+ γmt−1 + ωt)
Substituting these into (68), we have that the following condition must hold forall realizations of mt−1 and the random disturbances:
(1 + a)(1 + d2)pt = d2 (µ+ γmt−1 + ωt) + a(1 + d2) [b0 + b1mt−1]
+ [b0 + b1 (µ+ γmt−1 + ωt)]− d2vt + ut − (1 + d2)εt
= d2µ+ a(1 + d2)b0 + b0 + b1µ+ [d2γ + a(1 + d2)b1 + b1γ]mt−1
+ut − (1 + d2)εt − d2vt + (d2 + b1)ωt
Using the proposed solution for pt, this requires that
(1 + a)(1 + d2)b0 = d2µ+ a(1 + d2)b0 + b0 + b1µ
(1 + a)(1 + d2)b1 = d2γ + a(1 + d2)b1 + b1γ
(1 + a)(1 + d2)b2 = 1
(1 + a)(1 + d2)b3 = −(1 + d2)
(1 + a)(1 + d2)b4 = −d2
(1 + a)(1 + d2)b5 = d2 + b1
Solving these yields the following solution for pt:
pt = b0 + b1mt−1 + b2ut + b3εt + b4vt + b5ωt
pt =µ(1 + d2)
1− γ + d2+
(γd2
1− γ + d2
)mt−1
+ut − (1 + d2)εt − d2vt
(1 + a)(1 + d2)+
(d2
(1 + a)(1− γ + d2)
)ωt
40
Using this result for pt, the equilibrium expressions for output can be obtainedfrom (5.81) as
yt = a
[ut − (1 + d2)εt − d2vt
(1 + a)(1 + d2)+
(d2
(1 + a)(1− γ + d2)
)ωt
]+ εt
= a
[ut − d2vt
(1 + a)(1 + d2)+
(d2
(1 + a)(1− γ + d2)
)ωt
]+
(1
1 + a
)εt
while from (5.83) and (5.84),
it = d2 (yt −mt + pt + vt)
=µγd2
1− γ + d2+
(d2
1 + d2
)(ut + vt)−
(1− γ
1− γ + d2
)d2 (γmt−1 + ωt)
From these results, we can see that a positive realization of ω increases thecurrent price level and output and lowers the nominal rate of interest. Becauseoutput rises, the real rate of interest rt must fall to ensure aggregate demandand supply are equal at the temporarily higher level of output. When the moneysupply follows a random walk (γ = 1), a money supply shock has no effect onthe nominal interest rate, leading to a fall in the real rate but an equal rise inexpected inflation. When γ < 1, expected inflation rises less since the moneystock, after an initial increase, regresses back to its initial value.
To focus on the effect of ω on expected inflation, set u, v, and ε equal tozero, and use the equilibrium expression for pt to obtain
Etpt+1 − pt =
(γd2
1− γ + d2
)µ+
(γd2
1− γ + d2
)(γ − 1)mt−1
+
((1 + a)γ − 1
(1 + a)(1− γ + d2)
)d2ωt
while the impact on the real rate of interest is
rt = −yt = −(
a
(1 + a)(1− γ + d2)
)d2ωt
A rise in γ implies that ω has a larger impact on the expected future price levelsince it has a larger impact on the future money supply. This means the currentprice level rises more in response to a positive realizations of ω. In turn, thelarger unanticipated rise in pt generates a larger output increase and a largercorresponding fall in the real rate of interest. Since pt rises more when γ islarge, expected future inflation is less affected since the rise in the level of themoney supply and the price level is more persistent.
10. An increase in average inflation lowers the real demand for money. Demon-strate this by using the model given by equations (5.34) - (5.37) and as-suming the nominal money supply grows at a constant trend rate µ sothat mt = µt to show that real money balance mt − pt are decreasing inµ.
41
Answering this problem involves repeating the steps outlined in Section 5.7.3of the Chapter 5 Appendix, replacing the money supply process given in (5.86)with the one in the problem. Since the focus is on the effects of the deterministictrend µ on real money balances, it will simplify the problem if all stochasticdisturbances terms are ignored. In this case, the expression for the equilibriumprice level given in equation (5.85) becomes
pt =d2mt + a(1 + d2)Et−1pt +Etpt+1
(1 + a)(1 + d2)
=d2µt+ a(1 + d2)Et−1pt +Etpt+1
(1 + a)(1 + d2)(69)
For our guess for the solution, equation (5.87) is replaced by
pt = p0 + µpt
so that Et−1pt = p0 + µpt = pt and Etpt+1 = p0 + µp(t+ 1). Substituting theseinto (69),
pt =d2µt+ a(1 + d2)
(p0 + µpt
)+(p0 + µp(t+ 1)
)(1 + a)(1 + d2)
=[1 + a(1 + d2)] p0 + µp
(1 + a)(1 + d2)+
[d2µ+ a(1 + d2)µp + µp
(1 + a)(1 + d2)
]t
which equals p0 + µpt if and only if
µp =
[d2µ+ a(1 + d2)µp + µp
(1 + a)(1 + d2)
]⇒ µp = µ
and
p0 =[1 + a(1 + d2)] p0 + µp
(1 + a)(1 + d2)=
µ
d2
With this expression for the equilibrium price level, real money balances willequal
mt − pt = µt−(µ
d2+ µt
)= − µ
d2
which is decreasing in the growth rate of nominal money balances µ. From thesolution for the price level, the rate of inflation is equal to µ. Higher valuesof µ, and therefore higher rates of inflation, increase the opportunity cost ofholding money. This reduces the real demand for money, and, in equilibrium,real money balances are lower at higher rates of money growth.
42
5 Chapter 6: Money and the Open Economy
1. Suppose mt = m0 + γmt−1 and m∗t = m∗
0 + γ∗m∗t−1. Use equation (6.24)
to show how the behavior of the nominal exchange rate under flexibleprices depends on the degree of serial correlation exhibited by the homeand foreign money supplies.
Equation (6.24) on page 249 gives the following expression for the nominalexchange rate:
st =1
1 + δ
∞∑i=0
(δ
1 + δ
)i [(mt+i −m∗
t+i
)− (ct+i − c∗t+i
)]
Using the result that ct+i − c∗t+i = ct − c∗t , this becomes
st = − (ct − c∗t ) +1
1 + δ
∞∑i=0
(δ
1 + δ
)i [(mt+i −m∗
t+i
)](70)
which is equation (6.25) of the text. We now have to use the specified processes
for the nominal money supplies to evaluate the(
δ1+δ
)i (mt+i −m∗
t+i
)terms.
Since mt = m0 + γmt−1,
mt+1 = (m0 + γmt) = (1 + γ)m0 + γ2mt−1
and
mt+2 = m0(1 + γ + γ2) + γ3mt−1
Similarly,1
mt+i = m0
i∑j=0
γj + γi+1mt−1
= m01− γi+1
1− γ+ γi+1mt−1
For the foreign money supply,
m∗t+i =m∗
0
1− (γ∗)i+1
1− (γ∗)+ (γ∗)i+1
m∗t−1
1This uses the fact that 1 + a+ a2 + ...+ ak can be written as
(1 + a+ a2 + ...
)−
(ak+1 + ak+2 + ...
)=
(1 + a+ a2 + ...
)− ak+1
(1 + a+ a2 + ...
)
=1
1− a−
ak+1
1− a
for −1 < a < 1.
43
Now, for notational easy, let b =(
δ1+δ
). We need to evaluate
∑∞i=0 b
i[(mt+i −m∗
t+i
)]:
∞∑i=0
bi[(mt+i −m∗
t+i
)]=
∞∑i=0
bi
[m0
1− γi+1
1− γ−m∗
0
1− (γ∗)i+1
1− (γ∗)
]
+∞∑i=0
bi[γi+1mt−1 − (γ∗)i+1
m∗t−1
]Taking each term individually,
∞∑i=0
bi[m0
1− γi+1
1− γ
]=
m0
1− γ
∞∑i=0
bi[1− γi+1
]
=m0
1− γ
(1
1− b− γ
∞∑i=0
biγi
)
=m0
1− γ
(1
1− b− γ
1− bγ
)
∞∑i=0
bi
[m∗
0
1− (γ∗)i+1
1− (γ∗)
]=
m∗0
1− γ∗
(1
1− b− γ∗
1− bγ∗
)
∞∑i=0
biγi+1mt−1 = γmt−1
∞∑i=0
biγi =γmt−1
1− bγ
∞∑i=0
bi (γ∗)i+1m∗
t−1 = γ∗m∗t−1
∞∑i=0
bi (γ∗)i =γ∗m∗
t−1
1− bγ∗
Now collecting all these results, equation (70) becomes
st = − (ct − c∗t ) +1
1 + δ
{m0
1− γ
(1
1− b− γ
1− bγ
)− m∗
0
1− γ∗
(1
1− b− γ∗
1− bγ∗
)
+
(γmt−1
1− bγ− γ∗m∗
t−1
1− bγ∗
)}Since b = δ/(1−δ), 1/(1−b) is equal to 1+δ, and an expression like 1/(1−bγ)is equal to (1+ δ)/(1+ δ(1−γ)). So we can write the nominal exchange rate as
st = − (ct − c∗t ) +m0
1− γ
(1− γ
1 + δ(1− γ)
)− m∗
0
1− γ∗
(1− γ∗
1 + δ(1− γ∗)
)
+
(γmt−1
1 + δ(1− γ)− γ∗m∗
t−1
1 + δ(1− γ∗)
)
= − (ct − c∗t ) +m0
(1 + δ
1 + δ(1− γ)
)−m∗
0
(1 + δ
1 + δ(1− γ∗)
)
+
(γmt−1
1 + δ(1− γ)− γ∗m∗
t−1
1 + δ(1− γ∗)
)
44
To gain some insights from this expression, suppose that the money suppliesin both countries display the same autoregressive coefficient; γ = γ∗ and m0 =m∗
0. In this case,
st = − (ct − c∗t ) +
(γ
1 + δ(1− γ)
)(mt−1 −m∗
t−1
)The nominal exchange rate depends on the initial difference in the money sup-plies. If γ = 1, this difference is permanent (the money supplies follow randomwalks with drift), and st = − (ct − c∗t ) +
(mt−1 −m∗
t−1
), reflecting the perma-
nent difference associated with the difference in price levels when mt−1 �=m∗t−1.
If γ < 1 but positive, then both mt and m∗t follow stable processes that converge
to m0/(1 − γ). Any difference mt−1 − m∗t−1 is now transitory and so has a
smaller impact on the current nominal exchange rate.When γ �= γ∗, the comparison is more complicated, since the money supplies
in the two countries regress towards their steady-state values at different rates.The nominal exchange rate depends on the discounted value of the differencesin the paths followed by m and m∗.
2. In the model of Section 6.3 used to study policy coordination, aggregate-demand shocks were set equal to zero in order to focus on a commonaggregate-supply shock. Suppose instead that the aggregate supply shocksare zero, and the demand shocks are given by u ≡ x+ φ and u∗ ≡ x+ φ∗
so that x represents a common demand shock and φ and φ∗ are uncor-related country-specific demand shocks. Derive policy outcomes undercoordinated and (Nash) noncoordinated policy setting. Is there a role forpolicy coordination in the face of demand shocks? Explain.
Problems 2 - 5 use the model of section 6.3, so it will be convenient to developsome general expressions for output and inflation first, and then apply them tothe special cases considered in each of the problems. The basic model is givenin equations (6.35) - (6.39) on page 261 of the text. Equilibrium expressionsfor output in the two countries are given by equations (6.43) and (6.44) on page264.
Under a coordinated policy, the objective is to minimize
1
2E{λy2t + π2
t + λ(y∗t )2 + (π∗)2
}and the first order conditions for π and π∗ are (see page 265),
λb2A1yt + πt + λb2A2y∗t = 0 (71)
λb2A1y∗t + π∗
t + λb2A2yt = 0 (72)
45
It will prove convenient to use these to get expressions for πt−π∗t , the difference
in inflation between the two countries, and πt + π∗t , the sum of inflation in the
two countries. First subtracting and then adding (71) and (72),
πt − π∗t = −λb2(A1 −A2)(yt − y∗t )
πt + π∗t = −λb2(yt + y∗t )
where we have used the fact that A1 + A2 = 1 (see page 264). Since the termA1 −A2 will appear frequently, let H ≡ A1 −A2.
Now using (6.43) and (6.44) to find yt − y∗t and yt + y∗t , and substitutingthe results into the expressions for πt − π∗
t and πt + π∗t , we obtain
πt − π∗t = −λb2H [b2H(πt − π∗
t )
+H(et − e∗t ) + 2A3(ut − u∗t )]
=−λb2
[H2(et − e∗t ) + 2HA3(ut − u∗t )
]1 + λb22H
2(73)
πt + π∗t = −λb2 [b2(πt + π∗
t ) + et + e∗t ]
=−λb2 (et + e∗t )
1 + λb22(74)
For later reference, we can also derive
yt − y∗t = b2H(πt − π∗t ) +H(et − e∗t ) + 2A3(ut − u∗t )
= −b2H[λb2[H2(et − e∗t ) + 2HA3(ut − u∗t )
]1 + λb22H
2
]
+H(et − e∗t ) + 2A3(ut − u∗t )
=
(1
1 + λb22H2
)[H(et − e∗t ) + 2A3(ut − u∗t )] (75)
and
yt + y∗t = −b2[λb2 (et + e∗t )
1 + λb22
]+ et + e∗t =
[1
1 + λb22
](et + e∗t ) (76)
Note from these expressions that average inflation in the two countries re-sponds only to region-wide supply shocks (see equation 74), while inflation willrespond differently in the two countries to the extent that there are differentsupply shocks or different demand shocks (see equation 73).
Adding together (73) and (74) yields, in the cooperative equilibrium,
2πc,t = −λb2H [H(et − e∗t ) + 2A3(ut − u∗t )]
1 + λb22H2
− λb2 (et + e∗t )
1 + λb22
46
or2
πc,t = −1
2λb2
[H2(et − e∗t )
1 + λb22H2
+et + e∗t1 + λb22
]− λb2HA3(ut − u∗t )
1 + λb22(H)2
= −λb2[(
1 + λb22H2)et − 2A1A2(et − e∗t )
[1 + λb22H2] [1 + λb22]
+HA3(ut − u∗t )
1 + λb22H2
](77)
Then, from (74),
π∗c,t = −πc,t + et + e∗t
1 + λb22
= −λb2[(
1 + λb22H2)e∗t + 2A2A2(et − e∗t )
(1 + λb22H2) (1 + λb22)
− HA3(ut − u∗t )
1 + λb22H2
](78)
From (75) and (76),
yc,t =
(1
2
)(1
1 + λb22H2
)[H(et − e∗t ) + 2A3(ut − u∗t )]
+
(1
2
)[1
1 + λb22
](et + e∗t )
=
[A1
(1 + λb22H
2)et +A2
(1− λb22H
2)e∗t
(1 + λb22H2) (1 + λb22)
]+
(A3(ut − u∗t )
1 + λb22H2
)(79)
and
y∗c,t =
[A1
(1 + λb22H
2)e∗t +A2
(1− λb22H
2)et
(1 + λb22H2) (1 + λb22)
]−(A3(ut − u∗t )
1 + λb22H2
)(80)
These results all pertain to the case of a coordinated policy in the two coun-tries. Under policy without coordination, each country takes the inflation ratein the other as given in a Nash equilibrium. The home country’s policy makersets inflation to minimize E
(λy2t + π2
t
)while the foreign country policy maker
sets inflation to minimize E[λ(y∗t )
2 + (π∗)2]. The first order conditions take
the form
λb2A1yt + πt = 0
and
λb2A1y∗t + π∗
t = 0
It follows that
πt − π∗t = −λb2A1 (yt − y∗t )
2This uses the fact that (A1−A2)2 = (1−A2−A2)2 = (1− 2A2)2. Expanding the squareyields 1− 4A2(1−A2) = 1− 4A1A2.
47
and
πt + π∗t = −λb2A1 (yt + y∗t )
Using equations (6.43) and (6.44), these become
πt − π∗t = −λb2A1 [b2H(πt − π∗
t ) +H(et − e∗t ) + 2A3(ut − u∗t )]
= −λb2A1 [H(et − e∗t ) + 2A3(ut − u∗t )]
1 + λb22A1H(81)
and
πt + π∗t = −λb2A1 [−b2 (πt + π∗
t ) + et + e∗t ]
= −(
λb2A1
1 + λb22A1
)(et + e∗t ) (82)
Adding these together,
2πt = −[λb2A1 [H(et − e∗t ) + 2A3(ut − u∗t )]
1 + λb22A1H+
(λb2A1
1 + λb22A1
)(et + e∗t )
]
= −λb2A1
[2A1
(1 + λb22H
)et + 2A2e
∗t
[1 + λb22A1H] [1 + λb22A1]
]− 2λb2A1A3(ut − u∗t )
1 + λb22A1H
or
πN,t = −λb2A1
[A1
(1 + λb22H
)et +A2e
∗t
(1 + λb22A1H) (1 + λb22A1)+
(A3(ut − u∗t )
1 + λb22A1H
)](83)
From (82),
π∗N,t = −πN,t −
(λb2A1
1 + λb22A1
)(et + e∗t )
= −λb2A1
[A1
(1 + λb22H
)e∗t +A2et
(1 + λb22A1H) (1 + λb22A1)
−(A3(ut − u∗t )
1 + λb22A1H
)](84)
The sum and differences of output in the noncooperative equilibrium are
yt − y∗t =H(et − e∗t ) + 2A3(ut − u∗t )
1 + λb22A1H(85)
and
yt + y∗t =
(1
1 + λb22A1
)(et + e∗t ) (86)
48
Hence,
yN,t =
[A1
(1 + λb22H
)et +A2e
∗t
(1 + λb22A1H) (1 + λb22A1)
]+A3(ut − u∗t )
1 + λb22A1H(87)
and
y∗N,t =
(A1
(1 + λb22H
)e∗t +A2et
(1 + λb22A1H) (1 + λb22A1)
)− A3(ut − u∗t )
1 + λb22A1H(88)
We can now address the specific questions posed in Problem 2. For thisproblem, u ≡ x+φ and u∗ ≡ x+φ∗ so that x represents a common demand shockand φ and φ∗ are uncorrelated country-specific demand shocks, and et ≡ e∗t ≡ 0.thus, under a cooperative policy, equations (77), (78), (79), and (80) become
πc,t = −(
λb2HA3
1 + λb22H2
)(φt − φ∗t )
π∗c,t =
(λb2A3
1 + λb22H2
)(φt − φ∗t )
yc,t =
(A3
1 + λb22H2
)(φt − φ∗t )
and
y∗c,t = −(
A3
1 + λb22H2
)(φt − φ∗t )
From (83), (84), (87), and (88), equilibrium inflation rates and outputswithout cooperation are
πN,t = −(
λb2A1A3
1 + λb22A1H
)(φt − φ∗t )
π∗N,t =
(λb2A1A3
1 + λb22A1H
)(φt − φ∗t )
yN,t =
(A3
1 + λb22A1H
)(φt − φ∗t )
and
y∗N,t = −(
A3
1 + λb22A1H
)(φt − φ∗t )
49
Note that inflation response less to the relative demand shocks φt − φ∗t underthe coordinated policy than under the noncooperative policy as can be seen, forexample, by comparing the coefficients in the equilibrium expressions for πc,tand πN,t:
λb2HA3
1 + λb22H2<
λb2A1A3
1 + λb22A1H
(To see that the inequality follows, rewrite the comparison as
λb2HA3
[1 + λb22A1H
]< λb2A1A3
[1 + λb22H
2]
Dividing both sides by λb2A3,
H[1 + λb22A1H
]< A1
[1 + λb22H
2]
which becomes H < A1; recalling that H = A1 − A2, and both A1 and A2 arepositive, this inequality always holds.) This contrasts with the case consideredin the text; with only a common supply shock, inflation responds more under acoordinated policy (see page 267).
To investigate the potential role for policy coordination, we can evaluate thehome country’s loss function under the two policies. With coordination,
Lc =1
2
[λ
(A3
1 + λb22H
)2
+
(λb2HA3
1 + λb22H2
)2] (σ2φ + σ2
φ∗
)
=1
2
(λA2
3
1 + λb22H
)(σ2φ + σ2
φ∗
)(89)
In the noncooperative equilibrium,
LN =1
2
[λ
(A3
1 + λb22A1H
)2
+
(λb2A1A3
1 + λb22A1H
)2](σ2φ + σ2
φ∗
)
=1
2
[λA2
3
(1 + λb22A
21
)[1 + λb22A1H]
2
](σ2φ + σ2
φ∗
)(90)
Comparing (89) and (90), coordination yields a gain if and only if
LN > Lc
which occurs when
1 + λb22A21
[1 + λb22A1H]2 >
1
1 + λb22H
This comparison reduces to(1 + λb22H
) (1 + λb22A
21
)>[1 + λb22A1H
]2
50
Multiplying out both sides, this becomes
1 + λb22H + λb22A21 + λ2b42HA
21 > 1 + 2λb22A1H + λ2b42A
21H
2
or
H(1− 2A1) +A21 + λb22H(1−H)A2
1 > 0 (91)
But 1− 2A1 = (A1 +A2)− 2A1 = −(A1 −A2) = −H, so (91) becomes
−H2 +A21 + λb22H(1−H)A2
1 > 0
Since A21 −H2 = (A1 −H)(A1 +H) > 0 and 1−H > 0, the inequality holds.
Consider what happens in the face of a positive demand shock to the homecountry (φ > 0). The home country will deflate (π < 0) to partially offsetthe impact of the demand shock on domestic output. Because the home policyauthority takes foreign inflation as given in a Nash equilibrium, it expects thisdeflation to produce a real appreciation (see equation 6.42; ρ falls when π falls),reducing the impact of inflation on domestic output. More inflation volatility isneeded to maintain output stability. Thus, offseting demand shocks is perceivedto be more costly. With a coordinated policy, π is reduced while π∗ is increased,thus serving to stabilize output will smaller fluctuations in inflation.
3. Continuing with the same model as in the previous question, how are realinterest rates affected by a common aggregate-demand shock?
In the notation of Problem 2, x was a common aggregate demand shock.As was shown as part of the solution to Problem 2, neither home nor foreigninflation is affected by a common demand shock (equations 77, 78, 79, 80, 83,84, 87, and 88 all depended only on u−u∗ from which a common demand shockcancels out). With output and inflation independent of x, it must be that acommon demand shock alters real interest rates to offset the demand shock andmaintain aggregate demand constant (since output remains constant). Thus, apositive value of x should raise real interest rates in both countries; a negativex should lower real rates.
To verify this, first note from equation (6.42) on page 263 that a commondemand shock will leave the real exchange rate unchanged; the right side of (6.42)depends only on terms like π−π∗, e−e∗, and u−u∗, none of which are affectedby a common demand shock (see equations (73) and 81) With ρ unaffected, theinterest parity condition (6.39) implies that r∗ − r must remain unchanged, soboth interest rates change by the same amount. Adding together the aggregatedemand equations (6.37) and (6.38), we can obtain
rt + r∗t =
(1
a2
)[ut + u∗t − (1− a3)(yt + y∗t )]
51
If the x disturbance is the only shock, ut + u∗t = 2x and, from either (76) or(86), yt + y∗t is unaffected, so
rt + r∗t = 2
(1
a2
)x
Since we have seen that with only an x shock, r − r∗ = 0, it follows that
rt = r∗t =
(1
a2
)x
4. Policy coordination with asymmetric supply shocks: Continuing with thesame model as in the previous two questions, assume there are no demandshocks but that the supply shocks e and e∗ are uncorrelated. Derive policyoutcomes under coordinated and uncoordinated policy setting. Does co-ordination or noncoordination lead to greater or smaller inflation responseto supply shocks? Explain.
From equations (77), (78), (79), (80), (83), (84), (87), and (88) that werederived in the process of solving Problem 2, the outcomes under coordinated andnoncoordinated policies when u ≡ u∗ ≡ 0 are
πc,t = −λb2[(
1 + λb22H2)et − 2A1A2(et − e∗t )
[1 + λb22H2] [1 + λb22]
]
π∗c,t = −λb2
[(1 + λb22H
2)e∗t + 2A1A2(et − e∗t )
[1 + λb22H2] [1 + λb22]
]
yc,t =
[A1
(1 + λb22H
2)et +A2
(1− λb22H
2)e∗t
(1 + λb22H2) (1 + λb22)
]
y∗c,t =
[A1
(1 + λb22H
2)e∗t +A2
(1− λb22H
2)et
(1 + λb22H2) (1 + λb22)
]
πN,t = −λb2A1
[A1
(1 + λb22H
)et +A2e
∗t
(1 + λb22A1H) (1 + λb22A1)
]
π∗N,t = −λb2A1
[A1
(1 + λb22H
)e∗t +A2et
(1 + λb22A1H) (1 + λb22A1)
]
52
yN,t =
[A1
(1 + λb22H
)et +A2e
∗t
(1 + λb22A1H) (1 + λb22A1)
]
and
y∗N,t =
[A1
(1 + λb22H
)e∗t +A2et
(1 + λb22A1H) (1 + λb22A1)
]
Under a coordinated policy, the response to a domestic supply shock homeinflation is
πc,t = −λb2[
1 + λb22H2 − 2A1A2
(1 + λb22H2) (1 + λb22)
]et
while under noncoordinated policy it is
πN,t = −λb2[
A21
(1 + λb22H
)(1 + λb22A1H) (1 + λb22A1)
]et
These expressions are messy, so an easier way of comparing outcomes is toreturn to equation (73) and (74) for the case of cooperation and equations (81)and (82) for the noncooperation case. For the sum of inflation rates, (74) and(82) imply that πt+π∗
t responds more to et under cooperation that in the absenceof cooperation (the coefficient on e in 82 is increasing in A1 — which is less than1 — and under cooperation, the response in 74 is obtained by setting A1 = 1).So (πt + π∗
t )c responds more than (πt + π∗t )N . On the other hand, (πt − π∗
t )cresponds less than (πt − π∗
t )N as can be seen by comparing (73) and (81).3
The results that (πt + π∗t )c > (πt + π∗
t )N and (πt − π∗t )c < (πt − π∗
t )N allowsus to conclude that foreign inflation responds more (add the two inequalitiestogether), but it does not resolve whether domestic inflation responds more. Wesaw in the text that cooperation leads to a larger inflation response to a commonsupply shock than occurred without cooperation. Under cooperation in the faceof a home country supply shock, more of the adjustment is made by foreigninflation than would occur without cooperation, and this acts to allow domesticinflation to respond less. So inflation rate in the two countries diverge less( (πt − π∗
t )c < (πt − π∗t )N). But cooperation tends to lead to a stronger overall
response ( (πt + π∗t )c > (πt + π∗
t )N), so the net effect on πt is not clear.
5. Assume the home-country policy maker acts as a Stackelberg leader andrecognizes that foreign inflation will be given by equation (6.47). Howdoes this change in the nature of the strategic interaction affect the homecountry’s response to disturbances?
3This uses the fact that A1H > H2 since A1 > A2 (see the definitions of the A′
is on page
264).
53
Referring to section 6.3.3 of the text, the home policy authority picks in-flation to minimize λy2t + π2
t , but now, rather than taking foreign inflation π∗t
as given, the home policy authority recognizes that π∗t will be set according to
equation (6.47). Equation (6.47) was derived for the case of a single commonsupply shock; et = e∗t = εt. The first order condition for the home country willreflect this dependence of π∗
t on πt. Thus, the first order condition for the homecountry’s choice of inflation will be
λ
[b2A1 − b2A2
(λb22A1A2
1 + λb22A21
)]yt + πt = 0 (92)
This should be compared to the expression immediately above equation (6.46) on
page 266. the extra term −λb2A2
(λb22A1A2
1+λb22A21
)yt is the effect of that the home
country’s inflation rate has on home country output by causing foreign inflationto adjust based on the reaction function given by (6.47).
To solve (92), first note that the coefficient on yt can be rewritten as
b2A1 − b2A2
(λb22A1A2
1 + λb22A21
)= b2A1
[1 + λb22H
1 + λb22A21
]
since (A21 −A2
2) = H(A1 +A2) and A1 +A2 = 1. If we define
S ≡ 1 + λb22H
1 + λb22A21
< 1
then the first order condition for the home country becomes
λb2A1Syt + πt = 0
which contrasts with the first order conditon in the Nash case (λb2Ayt+πt = 0).One why to interpret this is that because S < 1, the marginal output efects of arise in home inflation are now smaller, since the home policy maker recognizesthat higher π will induce the foreign country to reduce its inflation rate, causinga depreciation for the home country ( ρ rises; see 6.42) that acts to offset theexpansionary impact of the rise in domestic inflation (see 6.35). It will beoptimal for π to response less. .
Also, using the foreign country’s reaction function, home country output(from 6.43) can be written
yt = b2A1πt − b2A2
[(λb22A1A2
1 + λb22A21
)πt +
(λb2A1
1 + λb22A21
)εt
]+ εt
= b2A1
[1 + λb22H
1 + λb22A21
]πt +
[1 + λb22A1H
1 + λb22A21
]εt
Now use the expression for home country output (equation 6.43 of the text) towrite the first order condition (92) as
λb2A1
[1 + λb22A1H
1 + λb22A21
]{b2A1
[1 + λb22H
1 + λb22A21
]πt +
[1 + λb22A1H
1 + λb22A21
]εt
}+ πt = 0
54
or
πt = −[
λb2A1
(1 + λb22A1H
)2(1 + λb22A
21S
s) (1 + λb22A21)
2
]εt
where
Ss ≡(1 + λb22A1H
) (1 + λb22H
)(1 + λb22A
21)
2
From (83), the response to a common supply shock in a Nash equilibrium is
πt = −[
λb2A1
(1 + λb22A1)
]εt
6. In a small open economy with perfectly flexible nominal wages, the textshowed that the real exchange rate and domestic CPI were given by
ρt =∞∑i=0
diEt
[a2r
∗t+i + et+i − ut+i
a1 + a2 + b1
]
and
pt =1
1 + c
∞∑i=0
(1
1 + c
)i
Et [mt+i − zt+i − vt+i]
where zt+i ≡ yt+i + (1 − h)ρt+i − crt+i. Assume r∗ = 0 for all t andthat e, u, and z + v all follow first order autoregressive processes (e.g.et = ρeet−1 + xet for xe white noise). Let the nominal money supply begiven by
mt = g1et−1 + g2ut−1 + g3 (zt−1 + vt−1)
Find equilibrium expressions for the real exchange rate, the nominal ex-change rate, and the consumer price index. What values of the parametersg1, g2, and g3 minimize fluctuations in st? in qt? in ρt? Are there anyconflicts between stabilizing the exchange rate (real or nominal) and sta-bilizing the consumer price index?
The first thing to note is that under the assumptions of the problem, the realexchange rate ρt is independent of the money supply process, depending only onthe exogenous behavior of r∗, et, and ut. Thus, in this example, the behavior ofρt is unaffected by the choice of the gi parameters. We can use the assumptionsof the problem to write the real exchange rate as
ρt =∞∑i=0
di[ρieet − ρiuuta1 + a2 + b1
]=
(1
a1 + a2 + b1
)[1
1− dρeet − 1
1− dρuut
](93)
55
To evaluate the expression for the equilibrium price pt, we do need to usethe money supply process:
pt =1
1 + c
∞∑i=0
(1
1 + c
)i
Et [g1et−1+i + g2ut−1+i + g3ωt−1+i − ωt+i]
=1
1 + c
∞∑i=0
(1
1 + c
)i [g1ρ
ieet−1 + g2ρ
iuut−1 + g3ρ
iωωt−1 − ρiω (ρωωt−1 + xωt)
]
where ωt ≡ zt − vt. Collecting terms,
pt =1
1 + c
∞∑i=0
(1
1 + c
)i [g1ρ
ieet−1 + g2ρ
iuut−1 + (g3 − ρω) ρ
iωωt−1 − ρiωxωt
]
=
(g1et−1
1 + c(1− ρe)
)+
(g2ut−1
1 + c(1− ρu)
)
+
((g3 − ρω)ωt−1
1 + c(1− ρω
)−(
xωt1 + c(1− ρω)
)(94)
So pt is stabilized if g1 = g2 = 0 and g3 = ρω. Since neither e nor u affect p inthis setup, any response by m to these disturbances would simply add additionalvariance to the price level. By setting g3 = ρω, policy is able to insulate pt fromthe forecastable movements in ωt.
The nominal exchange rate st is given by st = ρt − p∗ + pt where p∗ is theforeign price level. For simplicity, set p∗ = 0. Then st = ρt + pt. Combining(93) and (94), the nominal exchange rate is
st =
(1
a1 + a2 + b1
)[1
1− dρe(ρeet−1 + xet)− 1
1− dρu(ρuut−1 + xut)
]
+
(g1et−1
1 + c(1− ρe)
)+
(g2ut−1
1 + c(1− ρu)
)+
((g3 − ρω)ωt−1
1 + c(1− ρω
)−(
xωt1 + c(1− ρω)
)
The effects of ω shocks on the nominal exchange rate are minimized if g3 = ρω.The impact of et−1 is eliminated if(
1
a1 + a2 + b1
)(ρe
1− dρe
)+
(g1
1 + c(1− ρe)
)= 0
or
g1 = −(1 + c(1− ρe)
a1 + a2 + b1
)(ρe
1− dρe
)
The effects ut−1 on st can be eliminated if
−(
1
a1 + a2 + b1
)(ρu
1− dρu
)+
(g2
1 + c(1− ρu)
)= 0
56
or
g2 =
(1 + c(1− ρu)
a1 + a2 + b1
)(ρu
1− dρu
)
In terms of consumer prices qt, from equation (6.52) on page 270,
qt = hpt + (1− h)(st + p∗t )
= pt + (1− h)ρt (95)
Combining this with (93) and (94),
qt =
(g1et−1
1 + c(1− ρe)
)+
(g2ut−1
1 + c(1− ρu)
)+
((g3 − ρω)ωt−1
1 + c(1− ρω
)−(
xωt1 + c(1− ρω)
)
+
(1− h
a1 + a2 + b1
)[1
1− dρe(ρeet−1 + xet)− 1
1− dρu(ρuut−1 + xut)
]
The effects of ω shocks on the consumer price index are minimized if g3 = ρω.The impact of et−1 is eliminated if(
g11 + c(1− ρe)
)+
(1− h
a1 + a2 + b1
)(ρe
1− dρe
)= 0
or
g1 = −(1− h)
(1 + c(1− ρe)
a1 + a2 + b1
)(ρe
1− dρe
)
The effects ut−1 on qt can be eliminated if(g2
1 + c(1− ρu)
)−(
1− h
a1 + a2 + b1
)(ρu
1− dρu
)= 0
or
g2 = (1− h)
(1 + c(1− ρu)
a1 + a2 + b1
)(ρu
1− dρu
)
There are no conflicts (at least in this example) between stabilizing the nomi-nal exchange rate and stabilizing the consumer price level in the face of ω shocks( z − v). Since ω has no effect on the real exchange rate, stabilizing domesticprices would also stabilize the nominal exchange rate and the consumer pricelevel. For e and u disturbances, the appropriate responses of m to stabilize qare proportional to the optimal responses to stabilize s, but the responses aresmaller in absolute value by a factor 1−h if the objective is to stabilize q. Bothe and u affect the real exchange rate. Since qt = pt + (1 − h)ρt, policy canstabilize q by making p move to offset any movement in (1−h)ρ. The nominalexchange rate, however, is equal to ρt + pt, so it is stabilized it p fully adjuststo offset movements in ρ.
57
7. Equation (6.42) for the equilibrium real exchange rate in the two-countrymodel of section 6.3.1 takes the form ρt = AEtρt+1 + vt. Suppose vt =γvt−1 + ψt, where ψt is a mean-zero, white-noise process. Suppose thesolution for ρt is of the form ρt = bvt. Find the value of b. How does itdepend on γ?
First note that we can think of v as having a direct impact on ρ, holdingthe expected future real exchange rate constant, and an indirect effect if v altersEtρt+1. Under the proposed solution, Etρt+1 = bEtvt+1 = bγvt. Substitutingthis into the equilibrium condition for the real exchange rate,
ρt = Abγvt + vt = (1 +Abγ) vt
This can equal bvt for all realizations of vt if only if
b = (1 +Abγ)
from which it follows that
b =1
1−Aγ
The expression for b shows that a rise in γ increases b and leads an inno-vation in v to have a larger impact on the real exchange rate. This can be seenmore clearly by writing the solution for the real exchange rate as
ρt = bγvt−1 + bψt
If γ is large (i.e., close to 1 say), then innovations ψ to the v process are verypersistent. Therefore, in addition to the direct impact of v on ρ, there will be alarger impact on Etρt+1 if γ is large. Thus, the total impact of an innovationψt on the current real exchange rate ρt will be larger (i.e., b is larger).
58
6 Chapter 8: Discretionary Policy and Time In-
consistency
1. Assume firms maximize profits in competitive factor markets with laborthe only variable factor of production. Output is produced according tothe production function Y = ALα, 0 < α < 1. Labor is supplied inelas-ticly. Nominal wages are set at the start of the period at a level consistentwith market clearing, given expectations of the price level. Actual employ-ment is determined by firms once the actual price level is observed. Showthat, in log terms, output is given by y = αl∗ + α
1−α (p− pe)+ 11−α lnαA,
where l∗ is the log labor supply. (Note: the text contains a typo: al∗
appears instead of αl∗.)
Given the assumed form of the production function, the demand for laborcan be obtained from the condition that firms set the marginal product of labor,αALα−1, equal to the real wage W/P :
Ld =
(W
αAP
) −11−α
If L∗ is the fixed supply of labor, the equilibrium real wage that equates labordemand and labor supply is (
W
P
)∗= αA (L∗)α−1
or, in log terms,
(w − p)∗ = lnαA− (1− α)l∗
If workers and firms set the nominal wage at the start of the period, prior toobserving actual prices, then the contract wage consistent with labor marketclearing is
wc = Ep+ lnαA− (1− α)l∗
Actual (log) employment, given by the demand for labor, will be
l = − 1
1− α(wc − p− lnαA)
= − 1
1− α
(Ep− p− (1− α)l
)= l∗ +
1
1− α(p− Ep)
and the log of output is equal to
y = lnA+ αl
= lnA+α
1− α(p− Ep) + αl∗
59
2. Suppose an economy is characterized by the following three equations:
π = πe + ay + e
y = −br + u
∆m− π = −di+ y + v
where the first equation is an aggregate supply function written in theform of an expectations-augmented Phillips Curve, the second is an “IS” oraggregate-demand relationship, and the third is a money demand equationwhere ∆m denotes the growth rate of the nominal money supply. The realinterest rate is denoted by r and the nominal rate by i, with i = r + πe.Let the monetary authority implement policy by setting i to minimize theexpected value of 1
2
[λ(y − y∗)2 + π2
]where y∗ > 0. Assume the policy
authority has forecasts ef , uf , and vf of the shocks, but the public formsits expectations prior to the setting of i and without any information onthe shocks.
(a) Assume the monetary authority can commit to a policy of the formi = c0+c1e
f +c2uf +c3v
f prior to knowing any of the realizations ofthe shocks. Derive the optimal commitment policy (i.e., the optimalvalues of c0, c1, c2, and c3).
(b) Derive the time-consistent equilibrium under discretion. How doesthe nominal interest rate compare to the case under commitment?What is the average inflation rate?
(a) From the IS relationship, y = −b(i − πe) + u, so if we now use theaggregate supply relationship, π = πe − ab(i− πe) + au+ e, or
π = (1 + ab)πe − abi+ au+ e (96)
Taking expectations of both sides, conditional on the public’s information, πe =ie. Hence, inflation is
π = (1 + ab)ie − abi+ au+ e
and the objective function, expressed in terms of the policy instrument i becomes
L ≡ 1
2E{λ (−b(i− ie) + u− y∗)2 + [(1 + ab)ie − abi+ au+ e]2
}(97)
Under the commitment policy, the central bank follows a policy of the form
i = c0 + c1ef + c2u
f + c3vf (98)
60
where xf denotes the central bank’s forecast of x. With this policy rule, ie = c0.Substituting this result and the policy rule into (97) gives
L ≡ 1
2Eλ(−b(c1ef + c2u
f + c3vf ) + u− y∗
)2+1
2E[c0 − ab
(c1e
f + c2uf + c3v
f)+ au+ e
]2(99)
The objective is to minimizing this function by the choice of c0, c1, c2 andc3 where the choices are made prior to actually observing any of the shocks orforecasts. For c0, the first order condition is
E[c0 − ab
(c1e
f + c2uf + c3v
f)+ au+ e
]= c0 = 0
For c1, the first order condition (using c0 = 0) is
0 = E[λ(−b(c1ef + c2u
f + c3vf ) + u− y∗
)(−bef )]
+E[−ab (c1ef + c2u
f + c3vf)+ au+ e
](−abef )
If the shocks are mutually uncorrelated, this becomes
c1(λ+ a2)b2σfe − abσfe = 0
where σfe is the variance of the forecast of e and the result from rational expec-tations that E(eef ) = σfe has been used Hence,
c1 =a
(λ+ a2)b
For c2, one obtains
0 = E[λ(−b(c1ef + c2u
f + c3vf ) + u− y∗
)(−buf )]
+E[−ab (c1ef + c2u
f + c3vf)+ au+ e
](−abuf )
or
c2(λ+ a2)b2σfu − (λ+ a2)bσfu = 0
yielding
c2 =1
b
The first order condition for c3 is
0 = Eλ(−b(c1ef + c2u
f + c3vf ) + u− y∗
)(−bvf )
+E[−ab (c1ef + c2u
f + c3vf)+ au+ e
](−abvf ))
or
c3 = 0
61
Hence, the optimal commitment policy is
ic =1
b
(uf +
a
λ+ a2ef)
which does not depend on the money demand function at all.
(b) Under discretion, the central bank treats expectations as given in choosingi to minimize its expected loss function, based on its forecasts of the underlyingshocks. In this case, the expectations operator in the loss function (97) is con-ditional on ef , uf , and vf . The first order condition for the choice of i underdiscretion is
(−b)λ (−b(i− ie) + uf − y∗)− ab
[(1 + ab)ie − abi+ auf + ef
]= 0
or
λ(b2(i− ie)− buf + by∗
)− ab[(1 + ab)ie − abi+ auf + ef
]= 0
Solving for i,
i =(λb+ a(1 + ab)) ie − λy∗ + (λ+ a2)uf + aef
b(λ+ a2)(100)
Taking expectations based on the public’s information,
ie =λy∗
a
Hence, (100) becomes
i =λy∗
a+
1
b
(uf +
a
λ+ a2ef)
(101)
Notice that policy under discretion responds to the stochastic shocks the sameway as policy would if commitment were possible. However, the nominal rateunder discretion is systematically higher than under commitment. Since expectedinflation is equal to ie,
πe =λy∗
a> 0
3. Verify that the optimal commitment rule that minimizes the unconditionalexpected value of the loss function given by (8.10) is ∆mc = − aλ
1+a2λe.
62
The loss function (8.10) is
V c =1
2λ [a(b1e+ v) + e− k]
2+
1
2[b0 + b1e+ v]
2
Under the commitment policy, the central bank chooses b0 and b1 to minimizethe unconditional expectation of V c. If the shocks are uncorrelated, we can writethis expectation as
EV c =1
2λ[(1 + ab1)
2σ2e + a2σ2
v + k2]+
1
2
[b20 + b21σ
2e + σ2
v
]where σ2
x is the variance of x. If we minimize this with respect to b0 and b1,the first order condition for b0 is
b0 = 0
while that for b1 is
aλ(1 + ab1)σ2e + b1σ
2e = 0
or b1 = −aλ/(1 + a2λ). Hence, the optimal commitment policy is
∆mc = − aλ
1 + a2λe
as claimed.
4. Suppose the central bank acts under discretion to minimize the expectedvalue of equation (8.2). The central bank can observe e prior to setting∆m, but v is observed only after policy is set. Assume, however, that eand v are correlated, and that the expected value of v, conditional on e,is E [v|e] = qe where q = σv,e/σ
2e and σv,e is the covariance between e and
v.
(a) Find the optimal policy under discretion. Explain how policy de-pends on q.
(b) What is the equilibrium rate of inflation? Does it depend on q?
(a) The loss function (8.2) is quadratic in inflation and the deviation ofoutput from a target level. Taking the model to consist of equations (8.2) -(8.4), the central bank’s objective function can be written as
EV =1
2Eλ [a(π − πe) + e− k]2 +
1
2E [π]2
=1
2Eλ [a(∆m+ v −∆me) + e− k]2 +
1
2E [∆m+ v]2
63
If the central bank takes private sector expectations as given and can observe eprior to setting policy, then the first order condition for the optimal choice of∆m is
aλ [a(∆m+E [v|e]−∆me) + e− k] + ∆m+E [v|e] = 0
or
aλ [a(∆m+ qe−∆me) + e− k] + ∆m+ qe = 0
Hence,
∆m =a2λ∆me + aλk − (1 + q(1 + a2λ
)e
1 + a2λ
Taking expectations, conditional on the public’s information set,
∆me = aλk
so
∆m = aλk −(1 + q(1 + a2λ
)e
1 + a2λ(102)
The optimal respond to e will depend on q. If the central bank could observev, it would adjust ∆m to offset the impact of v on inflation. If e providessome information on which the central bank can base a forecast of v, then itwill adjust ∆m to offset the forecasted impact of v on inflation. To see this,note that (102) can be written as
∆m = aλk − 1
1 + a2λe− qe
= aλk − 1
1 + a2λe− E [v|e]
(b) The equilibrium rate of inflation is ∆m+ v or aλk − (1+q(1+a2λ)e1+a2λ + v.
this depends on q since the central bank’s choice of money growth is a functionof q. The average rate of inflation, or the inflation bias, is, however, equal toaλk, and this is independent of q.
5. Since the tax distortions of inflation are related to expected inflation,suppose the loss function (8.2) is replaced by
L = λ(y − yn − k)2 + (πe)2
where y = yn + a(π − πe). How is Figure 8.2 modified by this change inthe central bank’s loss function? Is there an equilibrium inflation rate?Explain.
64
The inflation loss is now assoicated with expected inflation. Since the centralbank is assumed, under discretion, to take expected inflation as given, it will nowview the costs of inflation as given, so increasing inflation a bit is perceived bythe central bank to yield benefit in terms of higher output but no cost.
To see this, substitute the aggregate supply relationship into the loss functionto obtain
L = λ [a(π − πe)− k]2 + (πe)2
If this is minimized with respect to π, the first order condition is
2aλ [a(π − πe)− k] = 0
or
π = πe +k
a2λ(103)
There is no expected rate of inflation such that πe = πe + k/a2λ. Figure 8.2,giving the central bank’s optimal choice of inflation, as a function of the public’sexpected rate of inflation, shows that the central bank’s reaction function givenby (103) does not cross the 45◦ line.
6. Based on Jonsson (1995) and Svensson (1997b). Suppose equation (8.3)is modified to incorporate persistence in the output process:
yt = (1− θ)yn + θyt−1 + a(πt − πet ) + et; 0 < θ < 1
Suppose the policy maker has a two-period horizon with objective functiongiven by
L = minE [Lt + δLt+1]
where Li =12
[λ(yi − yn − k)2 + π2
i
].
(a) Derive the optimal commitment policy.
(b) Derive the optimal policy under discretion without commitment.
(c) How does the presence of persistence (θ > 0) affect the inflation bias?
(a) Under commitment, the central bank follows a rule that specifies howinflation will be set as a function of the state of the economy. The rule is chosenbefore the central bank knows the current state of the economy, and the publicuses the rule to form their expectations about policy. They can do so since, byassumption, the central bank is committed to following the rule. Since the state
65
Figure 1: Dashed line is the central bank’s reaction function.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.05 0.1 0.15 0.2 0.25 0.3
Expected Inflation
Cen
tral
Ban
k's
Pla
nned
Infla
tion
Rat
e
Figure 2:
at time t is characterized by yt−1 and et, one can specify the commitment policyas
πt = b0 + b1xt−1 + b2et (104)
for period t, and
πt+1 = b′0 + b′1xt + b′2et+1 (105)
for period t+ 1 where
xt−1 ≡ yt−1 − yn
will be used to denote the output gap. Because the policy choice at time t may,through xt, affect output at time t+1, while any future effect of policy in t+1doesn’t matter (since the loss function only involves Lt and Lt+1), the optimalresponse to et may differ from the optimal response to et+1. For this reason,the coefficients in (104) and (105) are allowed to differ.
We need to find the optimal values of the coefficients in the policy rules thatminimize the unconditional value of the loss function L. Here, we can think ofthe central bank deciding on the parameters of the policy rule prior to havingany information about the state of the economy. Thus, it evaluates things fromthe perspective of the unconditional expectation of et (equal to zero) and xt−1
(also equal to zero)
66
The public knows the value of xt−1 when forming expectations of time t, sousing the policy rule (104),
πet = b0 + b1xt−1
and
πt − πet = b2et
Using the aggregate supply relationship,
xt = θxt−1 + (1 + ab2)et
Similarly, πt+1 − πet+1 = b′2et+1 and
xt+1 = θxt + (1 + ab′2)et+1
= θ2xt−1 + θ(1 + ab2)et + (1 + ab′2)et+1
Substituting these expressions into the loss function yields
E [Lt + δLt+1] =1
2E[λ [θxt−1 + (1 + ab2)et − k]2 + (b0 + b1xt−1 + b2et)
2]
+1
2δEλ
[θ2xt−1 + θ(1 + ab2)et + (1 + ab′2)et+1 − k
]2+1
2δE [b′0 + b′1 (θxt−1 + (1 + ab2)et) + b′2et+1]
2
Now minimize this with respect to the parameters in the policy rule. Doing soyields the following first order conditions:
For b0:
E [b0 + b1xt−1 + b2et] = 0
or
b0 = 0
For b1:
E [(b0 + b1xt−1 + b2et)xt−1] = 0
or
b1 = 0
For b2:
0 = Eλ [(θxt−1 + (1 + ab2)et − k)aet] + (b0 + b1xt−1 + b2et) et
+λδE[(θ2xt−1 + θ(1 + ab2)et + (1 + ab′2)et+1 − k
)θaet
]+λδE [(b′0 + b′1 (θxt−1 + (1 + ab2)et) + b′2et+1) b
′1aet]
67
or aλ(1+ab2)σ2e + b2σ
2e+λδθ2a(1+ab2)σ
2e +λδa (b′1)
2(1+ab2)σ
2e = 0. Solving
for b2,
b2 = − aλ+ aλδθ2 + aλδ (b′1)2
1 + a2λ+ a2λδθ2 + a2λδ (b′1)2
which depends on b′1. So turning to the first order condition for b′1,
δE [(b′0 + b′1 (θxt−1 + (1 + ab2)et) + b′2et+1) (θxt−1 + (1 + ab2)et)] = 0
or δb′1E (θxt−1 + (1 + ab2)et)2 = 0 which implies
b′1 = 0
Substituting this into the expression for b2,
b2 = −(
aλ(1 + δθ2
)1 + a2λ
(1 + δθ2
))
The first order condition for the optimal b′2 is
0 = δE[λ[θ2xt−1 + θ(1 + ab2)et + (1 + ab′2)et+1 − k
]aet+1
]+δE [(b′0 + b′1 (θxt−1 + (1 + ab2)et) + b′2et+1) et+1]
or
δaλ(1 + ab′2)σ2e + δb′2σ
2e = 0
which yields
b′2 = −(
aλ
1 + a2λ
)
Finally, it is straightforward to show that b′0 = 0. Thus, the optimal commitmentpolicy takes the form
πct = −(
aλ(1 + δθ2
)1 + a2λ
(1 + δθ2
))et (106)
and
πct+1 = −(
aλ
1 + a2λ
)et+1 (107)
Notice that average inflation is zero in each period. Also, the response toet+1 given by b′2 is exactly the same as obtained in, for example, equation (8.11)of the text. Since in period t+1 we have a standard one period problem, this isnot surprising. In period t, however, the optimal response to et differs from the
68
standard optimal response to a supply shock in a one-period model if δθ is non-zero. Assuming δ > 0 (the central bank cares about both period), the optimalcommitment policy in period t reduces to the standard result if θ = 0. If θ �= 0,then the optimal response to a period t supply shock as affected. Suppose θ > 0.Then it becomes optimal to offset more of the impact of et on period t outputby responding more strongly to et;
aλ(1 + δθ2
)1 + a2λ
(1 + δθ2
) > aλ
1 + a2λ
Since xt will affect xt+1, period t+ 1 output can be made more stable by insu-lating xt more from et.
(b) Under discretion, the central bank picks π in each period, taking expec-tations and the previous period’s output as given. Solving backwards, the centralbank will choose inflation in period t+ 1 to minimize
1
2λ[θxt + a(πt+1 − πet+1) + et+1 − k
]2+
1
2πt+1
taking xt and πet+1 as given. The first order condition is
aλ[θxt + a(πt+1 − πet+1) + et+1 − k
]+ πt+1 = 0
or
πt+1 =a2λπet+1 − aλ (θxt + et+1 − k)
1 + a2λ
Hence,
πet+1 = aλk − aλθxt
Expected inflation is decreasing in last period’s output if output displays positivepersistence ( θ > 0). A positive value of xt means that output in period t + 1will be closer to the desired level yn + k. Thus, the central bank’s incentive toinflate is reduced. Anticipating this, the public expects lower inflation.
Actual inflation in period t+ 1 under discretion will equal
πdt+1 = aλk − aλθxt −(
aλ
1 + a2λ
)et+1 (108)
and output will be
yt+1 = yn + θ(yt − yn) +
(1
1 + a2λ
)et+1
Notice that the response to et+1 is the same as under the optimal committmentpolicy given by equation (107).
69
The central bank’s loss in period t+ 1 is
Lt+1 =1
2λ
[θxt +
1
1 + a2λet+1 − k
]2+
1
2
[aλk − aλθxt − aλ
1 + a2λet+1
]2
Now use these results to evaluate the central bank’s decision problem in periodt. The two period loss function is
L = λE [xt − k]2 +Eπ2t +
1
2δλE
[θxt +
1
1 + a2λet+1 − k
]2
+1
2δE
[aλk − aλθxt − aλ
1 + a2λet+1
]2
Since xt = θxt−1 + a(πt − πet ) + et, the first order condition for the optimalinflation rate in period t, taking xt−1 and expectations as given, is
0 = aλ [xt − k] + πt + aθδλE
[θxt +
1
1 + a2λet+1 − k
]
−a2λθδE[aλk − aλθxt − aλ
1 + a2λet+1
]
or
0 = aλ [θxt−1 + a(πt − πet ) + et − k] + πt
+aθδλ [θ(θxt−1 + a(πt − πet ) + et)− k]
−a2λθδ [aλk − aλθ (θxt−1 + a(πt − πet ) + et)]
Solving for πt,
πt =
(a2λ(1 + δθ2 + aλθ2δ2
)1 + a2λ
(1 + δθ2 + a2λθ2δ
))πet +
(aλ(1 + δθ + a2λθδ
)1 + a2λ
(1 + δθ2 + a2λθ2δ
))k
−aλθ(
1 + δθ3 + a2λθ3δ
1 + a2λ(1 + δθ2 + a2λθ2δ
))xt−1 − aλ
(1 + δθ2 + a2λθ2δ
1 + a2λ(1 + δθ2 + a2λθ2δ
))et
It follows that
πet = aλ(1 + δθ + a2λθδ
)k − aλθ
(1 + δθ3 + a2λθ3δ
)xt−1
and inflation under discretion is equal to
πdt = aλ(1 + δθ(1 + a2λ)
)k − aλθ
(1 + δθ3 + a2λθ3δ
)xt−1
−aλ(
1 + δθ2(1 + a2λ)
1 + a2λ[1 + δθ2(1 + a2λ)
])et (109)
70
(c)Optimal discretionary policy is characterized by equations (109) for periodt and (108) for period t+1. In period t, the fact that output displays persistencedoes affect the inflation bias. Creating an output expansion in period t will tendto raise output in period t+ 1, so this increases the incentive to expand outputin period t. Anticipating this, the public expects higher inflation in period t, andaverage inflation equals aλ
(1 + δθ(1 + a2λ)
)k which exceeds the one-period bias
aλk. The inflation bias is increasing in θ since a larger θ implies that any outputexpansion in t will lead to a larger expansion in t+ 1. In addition, a low valueof yt−1 (so xt−1 is negative) acts to move yt further from the central bank’sdesired level yn−k; this increases the incentive to inflate and raises the expectedand actual rate of inflation. Comparing (109) and (106) shows that the period tresponse to et is also distorted under discretion. this is in contrast to the resultwe have usually found. When θ �= 0 , however, the response to et is larger, inabsolute value, under discretion
7. Show that πd given by (8.18) is equal to inflation under discretion when theweight on inflation in (8.2) becomes 1+δ and the economy is characterizedby (8.3) and (8.4).
The central bank’s decision problem under discretion can be written as
min
{1
2λ [a(π − πe) + e− k]2 +
1
2(1 + δ)π2
}
where π = ∆m + v and the central bank observes e but not v beforedetermining policy. The first order condition is
aλ [a(∆m−∆me) + e− k] + (1 + δ)∆m = 0
or
∆m =a2λ∆me − aλe+ aλk
1 + δ + a2λ
It follows that ∆me = aλk/(1 + δ). Hence
∆m =a2λ(aλk1+δ
)− aλe+ aλk
1 + δ + a2λ
=
(aλk
1 + δ
)− aλe
1 + δ + a2λ
and inflation is
π =
(aλk
1 + δ
)− aλe
1 + δ + a2λ+ v
as claimed in equation (8.18).
71
8. Suppose that the private sector forms expectations according to
πet = π∗ if πt−1 = πet−1
πet = aλk otherwise.
If the central bank’s objective function is given by (8.2) and its discountrate is β, what is the minimum value of π∗ that can be sustained inequilibrium?
The central bank’s single period loss function, from (8.2), is
Lt =1
2
[λ(yt − yn − k)2 + π2
t
]Assume output is given by equation (8.3) of the text:
yt = yn + a(πt − πet )
where the aggregate supply shock has been set to zero to parallel the analysis ofreputation in section 8.3.1.1. To determine the incentive the central bank hasto deviate from maintaining an inflation rate of π∗, we need to consider whathappens if the central bank deviates in period t, incurs the punishment in periodt+1 (the punishment being that the public expects an inflation rate of aλk) andthen returns to an inflation rate of π∗ in period t+2. Let Lnd
t be the loss fromno-deviation in period t and Ld
t the loss from deviating. The central bank willwish to deviate if
Ldt + βLd
t+1 < Lndt + βLnd
t+1
Following the discussion in the text, we can say that the central bank will havean incentive to deviate if the gain exceeds the cost, or
G(π∗) ≡ Lndt − Ld
t > β(Ldt+1 − Lnd
t+1
) ≡ C(π∗) (110)
To evaluate these terms, start with the loss under the no-deviation case.Not deviating means the central bank sets the rate of inflation equal to π∗ eachperiod. This is expected by the public, so output is equal to yn. The loss eachperiod is then
Lndt = Lnd
t+1 =1
2
[λk2 + (π∗)2
]If the central bank deviates in period t, then the public will expect an inflation
rate equal to the one-shot discretionary rate aλk in period t + 1. The best thecentral bank can do is to inflation at this rate, ensuring yt+1 = yn and
Ldt+1 =
1
2
[λk2 + (aλk)2
]=
1
2λ[1 + a2λ
]k2
72
Now we can determine Ldt . Since the public is expecting π
∗, the central bank,if it deviates, will pick πt to minimize
Ldt =
1
2
[λ(a(πt − π∗)− k)2 + π2
t
]The first order condition is
(1 + a2λ)πt − a2λπ∗ − aλk = 0
or
πt =a2λπ∗ + aλk
1 + a2λ
Output in period t will equal
yt = yn + a(πt − π∗)
= yn + a
(aλk − π∗
1 + a2λ
)
and
Ldt =
1
2
[λ
(a
(aλk − π∗
1 + a2λ
)− k
)2
+
(a2λπ∗ + aλk
1 + a2λ
)2]
=1
2
[λ
(k + aπ∗
1 + a2λ
)2
+
(a2λπ∗ + aλk
1 + a2λ
)2]
=1
2λ(k + aπ∗)2
1 + a2λ
We can now evaluate the gains and cost of deviation:
G(π∗) = Lndt − Ld
t
=1
2
[λk2 + (π∗)2
]− 1
2λ(k + aπ∗)2
1 + a2λ
and
C(π∗) = β(Ldt+1 − Lnd
t+1
)= β
{1
2λ[1 + a2λ
]k2 − 1
2
[λk2 + (π∗)2
]}
=1
2β[a2λ2k2 − (π∗)2
]The inflation rate π∗ can be sustained as an equilibrium if
G(π∗) < C(π∗)
73
or
1
2
[λk2 + (π∗)2
]− 1
2λ(k + aπ∗)2
1 + a2λ<
1
2β[a2λ2k2 − (π∗)2
]Simplifying, this condition becomes(
1 + β(1 + a2λ))(π∗)2 − 2aλkπ∗ +
[1− β
(1 + a2λ
)]a2λ2k2 < 0 (111)
The questions asks for the minimum value of π∗ that can be sustained asan equilibrium, given the trigger strategy followed by the public in forming theirexpectations. So we want to find the minimum value of π∗ that satisfies equation(111).
Solving the quadratic equation (111) for π∗,
π∗ =
2aλk ±√(2aλk)2 − 4
[1− β2 (1 + a2λ)2
]a2λ2k2
2 (1 + β(1 + a2λ))
Simplifying,
π∗ =aλk
[1± β
(1 + a2λ
)]1 + β(1 + a2λ)
so the minumum π∗ is given by
π∗min =
aλk[1− β
(1 + a2λ
)]1 + β(1 + a2λ)
Note that if 1 − β(1 + a2λ
)< 0, π∗
min < 0 and π∗ = 0 is a feasible equi-librium. This is the case shown in Figure 2 which plots G(π∗) − C(π∗) as afunction of π∗ for a = 1, λ = .5, β = .9 and k = .1.
9. Assume that nominal wages are set at the start of each period but thatwages are partially indexed against inflation. If wc is the contract basenominal wage, the actual nominal wage is w = wc + κ(pt − pt−1) where κis the indexation parameter. Show how indexation affects the equilibriumrate of inflation under pure discretion. What is the effect on averageinflation of an increase in κ? Explain why.
Referring to question 1, consider the same production set up so that employ-ment is again given (in log terms) by the marginal product condition:
l = − 1
1− α(w − p− lnαA)
74
Figure 3: Gain minus Cost of Deviating
-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
π∗π∗π∗π∗
G-C
Figure 4:
75
Substituting in the new expression for the nominal wage,
l = − 1
1− α(wc − (1− κ)p− κp−1 − lnαA)
If we assume the base contract wage wc is set equal to the expected marketclearing level, wc = Ep+ lnαA− (1−α)l∗ where l∗ is the fixed supply of labor.Actual employment is
l = − 1
1− α(Ep+ lnαA− (1− α)l∗ − (1− κ)p− κp−1 − lnαA)
= l∗ +1− κ
1− α(p− Ep)− κ
1− α(Ep− p−1)
while output is
y = αl∗ +α(1− κ)
1− α(p− Ep)− ακ
1− α(Ep− p−1) + e
= y∗ + a(1− κ) (p− Ep)− aκ (Ep− p−1) + e
where a = α/(1− α). Under discretion, Ep and p−1 are taken as givens whenthe central bank decides on its current price level. Thus, the important differenceintroduced by indexation is that the effect of a price surprise is decreasing in theindexation parameter κ.
If we now derive the optimal discretionary policy outcome, using for examplethe loss function (8.2) and repeating the analysis that led to equation (8.7), theparameter a of the text is replaced everywhere with a(1 − κ) and the inflationbias will be equal to
aλ(1− κ)k ≤ aλk
Notice that indexation ( 0 < κ < 1) reduces the average inflation bias. Byreducing the impact of a price surprise on real output, indexation reduces theincentive to induce an output expansion.
10. Suppose the central bank’s loss function is given by
V cb =1
2
[λ(y − yn − k)2 + (1 + δ)π2
)]If y = yn + a(π − πe) + e and π = ∆m+ v, verify that the inflation rateunder discretion is given by equation (8.18).
This problem actually is the same as number 7.
76
11. Beetsma and Jensen (1998): Suppose the social loss function is equal to
V s =1
2E[λ(y − yn − k)2 + π2
)]and the central bank’s loss function is given by
V cb =1
2E[(λ− θ)
(y − yn − k)2 + (1 + θ)
(π − πT
)2)]+ tπ
where θ is a mean zero stochastic shock to the central bank’s preferences,πT is an inflation target assigned by the government, and tπ is a linearinflation contract with t a parameter chosen by the government. Assumethat the private sector forms expectations before observing θ. Let y =yn+(π−πe)+e and π = ∆m+v. Finally, assume θ and the supply shocke are uncorrelated.
(a) Suppose the government only assigns an inflation target (so t = 0).What is the optimal value for πT ?
(b) Now suppose the government only assigns a linear inflation contract(so πT = 0). What is the optimal value for t?
(c) Is the expected social loss lower under the inflation target arrange-ment or the inflation contract arrangement?
(Notes: This statement of the problem corrects two typos in the text. Also,in the text, Beetsma and Jensen is listed as forthcoming. It has now appearedin the Journal of Money, Credit, and Banking, 30 (3), part 1, August 1998,384-403.)
(a) It will simplify to treat inflation as the central bank’s choice variable.From the link between money growth and inflation (π = ∆m + v), one caneasily obtain the rate of money growth needed to achieve the desired expectedinflation rate.
With only an inflation target, the central bank’s loss function is
V cb =1
2E[(λ− θ)(π − πe + e− k)2 + (1 + θ)
(π − πT
)2]where the relationship between output and surprise inflation has been used toeliminate y − yn. The first order condition for the optimal inflation setting,taking private expectations as given, is
(λ− θ)(π − πe + e− k) + (1 + θ)(π − πT
)= 0 (112)
Taking expectations conditional on the public’s information set (which does notinclude e or θ), −λk + πe − πT = 0 or
πe = πT + λk
77
Substituting this back into the central bank’s first order condition (112),
π = πT + (λ− θ)k − 1 + θ
1 + λe (113)
To find the optimal value for the inflation target, use (113) to evaluate socialloss, noting that π − πe = −θk − 1+θ
1+λe:
V s =1
2E
[λ
[−(1 + θ)k +
λ− θ
1 + λe
]2+
[πT + (λ− θ)k − 1 + θ
1 + λe
]2]
Minimizing this respect to the target inflation rate yields the first order condition
E
[πT + (λ− θ)k − 1 + θ
1 + λe
]= 0
or
πT = −λk
This is Svensson’s (1997) result: to offset the inflation bias, the inflation targetmust be set below the socially optimal inflation rate (equal to zero in this case). Ifthe inflation term in the social loss function had allowed for a non-zero optimalinflation rate, by having (π−π∗)2 rather than simply π2 in V s, then the optimaltarget would have been π∗ − λk.
(b) With a linear inflation contract but no inflation target, the central bank’sloss function is
V cb =1
2E[(λ− θ)(π − πe + e− k)2 + (1 + θ)π2
]+ tπ
and first order condition will be
(λ− θ)(π − πe + e− k) + (1 + θ)π + t = 0
Solving for inflation,
π =(λ− θ)(πe − e+ k)− t
1 + λ
The public will expect an inflation rate of λk − t. Hence, the central bank willdeliver an inflation rate of
π = (λ− θ)k − (1 + λ− θ)
1 + λt− 1 + θ
1 + λe (114)
Notice that the response to the supply shock is the same under either the target(see equation 113) or the contract (see 114).
78
To find the optimal value of t from the government’s perspective, evaluate
V s, first noting that equation (114) implies π − πe = −θ(k − 1
1+λ t)− 1+θ
1+λe:
V s =1
2Eλ
[−θ(k − 1
1 + λt
)+λ+ θ
1 + λe− k
]2
+1
2E
[(λ− θ)k − (1 + λ− θ)
1 + λt− 1 + θ
1 + λe
]2
The first order condition for the optimal t is
0 = Eλ
(θ
1 + λ
)[−θ(k − 1
1 + λt
)+λ+ θ
1 + λe− k
]
−E
((1 + λ− θ)
1 + λ
)[(λ− θ)k − (1 + λ− θ)
1 + λt− 1 + θ
1 + λe
]
Evaluating the expectations, and making use of the assumption that E(θe) = 0,
λ
(σ2θ
1 + λ
)(1
1 + λt− k
)− λk + t−
(σ2θ
1 + λ
)[k − 1
1 + λt
]= 0
Combining terms, this can be written as[1 +
(σ2θ
1 + λ
)]t− [λ+ σ2
θ
]k = 0
or
t =(1 + λ)
(λ+ σ2
θ
)1 + λ+ σ2
θ
k (115)
(c) To evaluate the social loss function under the alternative policies, it isuseful to start with the expressions for y− yn−k and π and their variances foreach type of policy. Let subscripts T and c denote the targeting regime and thecontract regime. Then, for the targeting regime,
πT = −θk − λ− θ
1 + λe⇒ Eπ2 = σ2
θk2 +
λ2 + σ2θ
(1 + λ)2σ2e
and
yT − yn − k = −(1 + θ)k +1 + θ
1 + λe
which implies
E(yT − yn − k)2 = (1 + σ2θ)k
2 +1 + σ2
θ
(1 + λ)2σ2e
79
For the contract regime, substitute (115) into (114) to obtain
πc = (λ− θ)k − (1 + λ− θ)
1 + λ
[(1 + λ)
(λ+ σ2
θ
)1 + λ+ σ2
θ
k
]− 1 + θ
1 + λe
=
[− θ + σ2
θ
1 + λ+ σ2θ
k
]− 1 + θ
1 + λe⇒ Eπ2 =
σ2θ(1 + σ2
θ)
(1 + λ+ σ2θ)
2k2 +
λ2 + σ2θ
(1 + λ)2σ2e
yc − yn − k = −(1 +
θ
1 + λ+ σ2θ
)k +
1 + θ
1 + λe
which implies
E(yc − yn − k)2 =
[1 +
σ2θ
(1 + λ+ σ2θ)
2
]k2 +
1 + σ2θ
(1 + λ)2σ2e
We can now evaluate social loss under the two policy regimes. For inflationtargeting,
V sT =
1
2λ
{(1 + σ2
θ)k2 +
1 + σ2θ
(1 + λ)2σ2e
}+
1
2
[σ2θk
2 +λ2 + σ2
θ
(1 + λ)2σ2e
]
=1
2
[λ(1 + σ2
θ) + σ2θ
]k2 +
1
2
[λ+ σ2
θ
(1 + λ)
]σ2e
while for the inflation contract regime,
V sc =
1
2λ
{[1 +
σ2θ
(1 + λ+ σ2θ)
2
]k2 +
1 + σ2θ
(1 + λ)2σ2e
}
+1
2
[σ2θ(1 + σ2
θ)
(1 + λ+ σ2θ)
2k2 +
λ2 + σ2θ
(1 + λ)2σ2e
]
=1
2
[λ+
σ2θ
1 + λ+ σ2θ
]k2 +
1
2
[λ+ σ2
θ
(1 + λ)
]σ2e
Subtracting V sc from V s
T yields
V sT − V s
c =1
2k2{[λ+ (1 + λ)σ2
θ
]− [λ+σ2θ
1 + λ+ σ2θ
]}
=1
2k2σ2
θ
{(1 + λ)−
[1
1 + λ+ σ2θ
]}
which is always positive if σ2θ > 0, i.e., if there is any uncertainty about the
central bank’s preferences. Thus, in the face of preference uncertainty, the linearinflation contract performs better than a simple inflation target.
80
7 Chapter 9: Monetary-Policy Operating Pro-
cedures
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). Assumethe central bank’s loss function is E[y]2.
(a) Under a money supply operating procedure, derive the value of mt
that minimizes E[y]2.
(b) Under an interest rate operating procedure, derive the value of itthat minimizes E[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 interestrate procedure depend on the ρ′is? Explain.
(e) Suppose the central bank sets its instrument for two periods (forexample, 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 theρ′is?
a) Under a money supply procedure, the money demand relationship impliesthat the interest rate is it = c−1(yt + vt −mt). Output is then equal to
yt = −αc(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−1 and E (ut) = ρuut−1, so this first order condition requires that mt satisfy
αmt − αρvvt−1 + cρuut−1 = 0
or
mt = ρvvt−1 −( cα
)ρuut−1
81
The money supply is adjusted to offset the forecasted effects of vt and ut onoutput:
yt =cϕt − αψt
c+ α(117)
b) Under an interest rate procedure, output is equal to
yt = −αit + ut (118)
and it is chosen to minimize E[y]2 = E[−αit + ut]2. The first order conditionis
−2αE[−αit + ut] = 0
or
it =ρuut−1
α
The interest rate is adjusted to offset the predicted aggregate demand shock, whilemoney demand shocks do not affect output and so do not require any adjustmentin the interest rate instrument.
c) As noted under parts (a) and (b), the optimal policy will involve trying toinsulate output from the two shocks ut and vt. If these could be observed beforepolicy is set, the optimal policies would be mt = vt −
(cα
)ut under a money
procedure and it =(1α
)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 ut and vt with 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 −
(cα
)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 betweena money procedure and an interest rate procedure will not depend on either ρu
82
or ρv. Since the predictable component of the shocks (the component that doesdepend on ρu and ρv) is offset under both policies, the comparison will onlydepend on how well the different policies insulate output from the unforecastableshocks (ϕt and ψt).
e) If the objective is to set m or i at time t for two period to minimizeE[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 serialcorrelation properties of the shocks. Starting with the money supply procedure,we can use (116) for output to write the loss function as
E
[α (m∗ − vt) + cut
c+ α
]2+ βE
[α (m∗ − vt+1) + cut+1
c+ α
]2since, by assumption, m is fixed for two periods. The first order condition is
2α
c+ α
{E
[α (m∗ − vt) + cut
c+ α
]+ βE
[α (m∗ − vt+1) + cut+1
c+ α
]}= 0
From the process followed by the disturbances, E(ut) = ρuut−1 and E(ut+1) =ρ2uut−1, while E(vt) = ρvvt−1 and E(vt+1) = ρ2vvt−1. Using these in the firstorder condition, the optimal m∗ must satisfy
α (m∗ − ρvvt−1) + cρuut−1 + β[α(m∗ − ρ2vvt−1
)+ cρ2uut−1
]= 0
or
m∗ =α(1 + βρv)ρvvt−1 − c(1 + βρu)ρuut−1
α(1 + β)
Since m is fixed for two periods, it adjusts to offset what amounts to the averagediscounted expected shocks over the two periods. As a consequence, output willnot be perfectly insulated from the forecasted components of ut, ut+1, vt, orvt+1:
yt =c[ut −
(1+βρu1+β
)ρuut−1
]− α
[vt −
(1+βρv
1+β
)ρvvt−1
]c+ α
=c [(1 + β)ϕt + β (1− ρu) ρuut−1]− α [(1 + β)ψt + β (1− ρv) ρvvt−1]
(1 + β) (c+ α)
=cϕt − αψt
c+ α+
(β
1 + β
)[c(1− ρu)ρuut−1 − α(1− ρv)ρvvt−1
c+ α
](119)
(which should be compared with equation 117) and
yt+1 =c[ut+1 −
(1+βρu
1+β
)ρuut−1
]− α
[vt+1 −
(1+βρv1+β
)ρvvt−1
]c+ α
=c[(1 + β)
(ϕt+1 + ρuϕt
)− (1− ρu) ρuut−1
](1 + β) (c+ α)
−α[(1 + β)
(ψt+1 + ρvψt
)− (1− ρv) ρvvt−1
](1 + β) (c+ α)
83
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 affectoutput 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 + βρu1 + β
)ρuut−1 = ϕt +
β(1− ρu)
1 + βρuut−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 ρuand ρu, the comparison of the loss functions under the two policies will nolonger be independent of the serial correlation properties of ut and vt. Forexample, suppose ρu = 0 but ρv �= 0. Under an interest rate policy, output doesnot depend on the v disturbances, so yt = ϕt and yt+1 = ϕt+1, but under themoney supply rule, equation (119) becomes
yt =cϕt − αψt
c+ α−(
β
1 + β
)(α
c+ α
)(1− ρv)ρvvt−1
84
and
yt+1 =
(cϕt+1 − αψt+1
c+ α
)−(
α
c+ α
)ρvψt +
α (1− ρv) ρvvt−1
(1 + β) (c+ α)
which still depends on ρv. Since σ2v = σ2
ψ/(1−ρ2v), the variance of output undera money supply procedure is
E[yt]2m =
(c
c+ α
)2
σ2ϕ +
(α
c+ α
)2[1 + ρ2v +
(1− ρv)2ρ2v
(1 + β)2(1− ρ2v)
]σ2ψ
=
(c
c+ α
)2
σ2ϕ +
(α
c+ α
)2[1 + ρ2v +
(1− ρv) ρ2v
(1 + β)2 (1 + ρv)
]σ2ψ
while under an interest rate procedure,
E[yt]2i = σ2
ϕ
2. Solve for the δ′is appearing in (9.11) and show that the optimal rule forthe base is the same as that implies by the value of µ∗ given in (9.10).
The δ′is that appear in equation (9.11) are obtained by calculating the leastsquares 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 rateis 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 v, ω, and u (see page 394).From the least squares formula, the forecast of a variable y, conditional on x,
is y =(σx,y
σ2x
)x where σx,y is the covariance between x and y and σ2
x is the
variance of x. (This assumes both y and x have zero means.)Applying this formula, we have
v =
(σv,iσ2i
)i
=
σ2
v
α+c+µ+h
σ2v+σ2
ω+σ2u
(α+c+µ+h)2
i = (α+ c+ µ+ h)
(σ2v
σ2v + σ2
ω + σ2u
)i
so
δv =
(α+ c+ µ+ h
σ2v + σ2
ω + σ2u
)σ2v
85
Similarly,
δω = −(α+ c+ µ+ h
σ2v + σ2
ω + σ2u
)σ2ω
and
δu =
(α+ c+ µ+ h
σ2v + σ2
ω + σ2u
)σ2u
The next step is to substitute these expressions into the policy rule (9.11):
b =
(−c+ h
αδu + δv − δω
)i
=
(α+ c+ µ+ h
σ2v + σ2
ω + σ2u
)(−c+ h
ασ2u + σ2
v + σ2ω
)i
or, since b = µi,
µ =
(α+ c+ µ+ h
σ2v + σ2
ω + σ2u
)(−c+ h
ασ2u + σ2
v + σ2ω
)
Solving this for µ yields
µ = − (c+ h) + α
(σ2v + ασ2
ω
σ2u
)
which proves that the policy rule (9.11) yields the same response to the interestrate as was found in equation (9.9), and the optimal policy rule is
b =
[− (c+ h) + α
(σ2v + ασ2
ω
σ2u
)]i = µ∗i
3. Suppose the money demand relationship is given by m = −c1i+ c2y +v.Show how the choice of an interest rate versus a money supply operatingprocedure 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 supplyprocedure are given by
i =
(1
c1
)(c2y −m+ v)
and
y = −(α
c1
)(c2y −m+ v) + u
=
(1
c1 + αc2
)[a(m− v) + c1u]
86
and the loss function E(y)2 is minimized when
2
(α
c1 + αc2
)E
[α(m− v) + c1u
c1 + αc2
]= 2
(α
c1 + αc2
)E
[αm
c1 + αc2
]= 0
or m = 0. The loss function is then equal to
L(m) =
(1
c1 + αc2
)2 (α2σ2
v + c21σ2u
)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 >
(c2 +
2c1α
)c2σ
2u
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 theimpact on output of a positive u shock. If c2 is large, the resulting rise inoutput has a large impact on money demand. This in turn causes interest ratesto rise, offsetting the original rise in output. Thus, u shocks have a smallerimpact on output under an m procedure when c2 is large. Similarly, a positivev 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 inlowering money demand. As a result, interest rates need to rise less to maintainmoney market equilibrium after the positive v shock, and, with a smaller rise ini, y falls less.
These results can be illustrated by Figure 1. The negatively sloped solid lineis the IS equation y = −αi when u takes on its expected value of 0. Thepositively 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 =(c2c1
)y).
The line BB is drawn for a larger value of c2. The dotted negatively sloped lineshows the results of a positive u shock; output rises less when c2 is large; outputrises from he the level associated with point C to D if c2 is small, but it risesonly to E when c2 is larger. A positive v shock shifts AA and BB to A′A′ and
B′B′ (since i =(
1c1
)(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 islarge, rather than C to G.
87
Figure 5: Chapter 9, Problem 3 — The impact of c2 under a money supplyoperating procedure
Output
Inte
rest
Rat
e
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 byusing the following model:
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 andthe policy instrument must be set prior to observing the current values ofthe disturbances.
(a) Calculate the expected loss function if it is used as the policy instru-ment. (Hint: Give the objective function, the instrument will alwaysbe 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
88
i. the relative variances of the aggregate supply, demand, andmoneydemand 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 equalπt =
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 observingthe disturbances, so in evaluating the first order condition, E (ut) = E(ut) =E (et) = 0. Thus, (100) becomes −αλit − α 1
a it = 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 − eta
Notice that under a policy that sets i = 0, inflation is a mean zero, seriallyuncorrelated 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)σ2
u + σ2e
a2(125)
b) Under a money rule, we need to use equation (122), solving it for thenominal 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
89
the rate of inflation while (122) involves the price level, we can either expressinflation as pt−pt−1, and then solve for the price level, or, since pt−1 is knownwhen policy is set, we could replace mt − pt in (122) with mt + pt−1 − πt andsolve for the rate of inflation. Since the loss function is expressed in terms ofinflation, 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 isthen
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 − yn and can beviewed as the policy instrument. The first order condition for the choice of µtis (
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 + αeta (c+ α) + α
90
and
πt =cut − αvt − (c+ α) et
a (c+ α) + α
and the loss function is
L(m) =
(1 + a2λ
) (c2σ2
u + 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) givenin (125) and the loss L(m) given in (129), with an interest rate instrumentpreferred if
L(i) < L(m)
or if
(1 + a2λ)σ2u + σ2
e
a2<
(1 + a2λ
) (c2σ2
u + 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]
a2α2σ2u +
[2a (c+ α) + α(1− a2λ)
](1 + a2λ)
σ2e
This shows that the comparison depends on the different variance terms. Amoney oriented operating procedure is less likely to be desirable if money demandshocks are large (i.e., σ2
v is large). The coefficient on σ2u is positive, so an
interest rate rule is more likely to be preferred if aggregate demand shocks areimportant (i.e., σ2
u is large), while it is also likely to be preferred if supplydisturbances are large (i.e., σ2
e 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 operatingprocedure comparison will be independent of λ. When aggregate supply shocksare present (σ2
e > 0), then there can be conflicts between stabilizing output andstabilizing inflation. If output objectives are very important (λ large), then itis 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 aninterest rate procedure, aggregate demand remains constant (see equation 124),so output is stabilized and inflation must fall. Such a policy will be preferredif λ is large. Under a money supply procedure, in comparison, output will riseand inflation will fall. Since both adjust, inflation falls by less than under the ipolicy. A policy maker who cares more about inflation stabilization (i.e., has alower λ) will prefer money supply operating procedure.
91
5. Using the intermediate target model of section (9.3.3) and the loss function(9.15), rank the policies that set it equal to ıt, i
Tt , and ιt + 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 = ıt = π∗ +
(1
α
)(ρuut−1 − ρzzt−1)
(see equation 9.17, page 397). The value of the loss function under this policywas given at the bottom of page 397:
V (ıt) =
(1
a
)2 (σ2ϕ + σ2
e
)(131)
Under the second policy,
it = iTt = ıt +(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 ofpage 400:
V (iTt ) =
(1
ac+ α(1 + a)
)2 (c2σ2
ϕ + (α+ c)2σ2e + α2σ2
ψ
)(132)
Comparing (131) and (132),
V (ıt)− V (iTt ) =
(1
a
)2 (σ2ϕ + σ2
e
)−( 1
ac+ α(1 + a)
)2 (c2σ2
ϕ + (α+ c)2σ2e + α2σ2
ψ
)
=
[[2acα(1 + a) + α2(1 + a)2
]σ2ϕ + 2αa2(c+ α)σ2
e
a2 (ac+ α(1 + a))2
]
−(
1
ac+ α(1 + a)
)2
α2σ2ψ
92
The first term is positive, indicating that the intermediate targeting rule leads toa smaller loss (V (ıt) > V (iTt )) if the only disturbances are demand and supplyshocks (ϕ and e). As discussed in the text, however, the intermediate targetingprocedure can do worse if money demand shocks are important (V (ıt) < V (iTt )if σ2
ψ is large).The final policy sets it equal to ιt+µ∗xt, where xt is defined below equation
(9.22) on page 401 as
xt =
(1 +
1
a
)ϕt −
(1
a
)et + ψt
and µ∗ was defined, also on page 401, as
µ∗ =
(1
α
)[a(1 + a)σ2
ϕ + aσ2e
(1 + a)2σ2ϕ + σ2
e + a2σ2ψ
]
Using the definition of ιt, the rate of interest under this policy is
it = ιt + µ∗xt = π∗ +
(1
α
)(ρuut−1 − ρzzt−1) + µ∗
[(1 +
1
a
)ϕt −
(1
a
)et + ψt
]
To evaluate the loss function under this policy, use equation (9.16) of thetext to find the equilibrium rate of inflation when it = ιt + µ∗xt:
π(ιt + µ∗xt) =(a+ α)π∗ − α(ιt + µ∗xt) + ut − zt
a
= π∗ +(ut − ρuut−1)− (zt − ρzzt−1)− αµ∗xt
a
= π∗ +
(1
a
)(ϕt − et − αµ∗xt)
Problem 2 showed that the value of µ∗ was related to the best forecasts of ϕ ande, conditional on observing x. We can write inflation under this policy as
π(ιt + µ∗xt) = π∗ +
(1
a
){[ϕt −E(ϕt | x)]− [et −E(et | x)]}
Comparing this to the policy that lead to V (ıt), in which π−π∗ =(1a
)(ϕt − et),
it is clear that the variance of inflation around π∗ will be smaller with theιt + µ∗xt policy since the variance of [ϕt −E(ϕt− | x)] is less than or equalto the variance of ϕ, and similarly for the comparison of the variances of[et −E(et | x)] and et. Therefore,
V ∗ ≡ V (ιt + µ∗xt) ≤ V (ıt)
To compare the loss under the intermediate target policy iT and the policythat optimal uses information ( ιt + µ∗xt), use the equation near the bottom ofpage 400 to write
iTt = ιt + µTxt
93
Using equation (9.16). inflstion can be written as
πt(iTt ) =
(a+ α)π∗ − α(ιt + µTxt
)+ ut − zt
a
= π∗ +ϕt − et − αµTxt
a
so the loss function V (iT ) is equal to the variance of(1a
) (ϕt − et − αµTxt
).
Inflation around π∗ under the ιt + µ∗xt policy is
πt(ιt + µ∗xt) = π∗ +ϕt − et − αµTxt
a
and the loss function V ∗ is the variance of(1a
)(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 according to (9.17), the ex-pected value of the nominal money supply is equal to m given in (9.19).
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 demandequation (9.14) can be written as
mt − pt−1 = πt + a (πt − π∗) + zt − cit + vt
= π∗ + (1 + a)
(ϕt − et
a
)+ zt − cπ∗ −
( cα
)(ρuut−1 − ρzzt−1) + vt
since E t−1πt = π∗. Taking expectations as of time t − 1 of this equation andsolving for E t−1mt,
Et−1mt = (1− c)π∗ + pt−1 −( cα
)ρuut−1 +
(1 +
c
α
)ρzzt−1 + ρvvt−1
which is the same as the expression for m in equation (9.19).
7. Suppose the central bank is concerned with minimizing the expected valueof a loss function of the form
L = E[TR]2 + χE[if ]2
which depends on the variances of innovations to total reserves and thefunds rate (χ is a positive parameter). Using the reserve market model ofsection 9.4.2, find the values of φd and φb that minimize this loss function.Are there conditions under which a pure nonborrowed reserves or a pureborrowed reserves operating procedures would be optimal?
94
The reserves market model from section 9.4.2 consists of a total reservesdemand equation, a borrowed reserves demand equation, a supply of nonborrowedreserves equation, and an equilibrium condition. These are specified as
TR = −αif + vd
BR = b(if − id) + vb
NBR = φdvd + φbvb + vs
and the equilibrium condition that
TR = BR+NBR
= αif + b(if − id) + (φd − 1)vd + (1 + φb)vb + vs
Substituting these first three equations into the equilibrium condition andsolving for the funds rate if yield
if =
(1
a+ b
)(bid − (1 + φb)vb + (1− φd)vd − vs
)
If the objective is to pick φd and φb to minimize L = E[TR]2 +χE[if ]2, thefirst order conditions will be
−2αE[−αif + vd]
(∂if
∂φd
)+ 2χE[if ]
(∂if
∂φd
)= 0 (133)
and
−2αE[−αif + vd]
(∂if
∂φb
)+ 2χE[if ]
(∂if
∂φb
)= 0 (134)
To evaluate these, note that(
∂if
∂φd
)= −vd/(a+ b), while
(∂if
∂φb
)= −vb/(a+ b).
Hence, equation (133) implies
0 = αE[αif − vd]
( −vda+ b
)+ 2χE[if ]
( −vda+ b
)
= −α2
(1− φd
(a+ b)2
)σ2d +
(α
a+ b
)σ2d − χ
(1− φd
(a+ b)2
)σ2d
Solving for φd,
(1− φd) =α(a+ b)
α2 + χ
95
φd = 1− α(a+ b)
α2 + χ(135)
Equation (134) yields
−αE(α(1 + φb)vb
a+ b
)( −vba+ b
)+ χE
(−(1 + φb)vb
a+ b
)( −vba+ 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 shockto borrowed reserve demand should generate an equal but opposite movementin nonborrowed reserves. This keeps total reserves unchanged. Since neithertotal reserve supply or demand have changed, the funds rate is left unchanged.Thus, setting φb = −1 and accommodating shifts in borrowed reserve demandcompletely insulates both TR and if from vb shocks.
From (135), the optimal value of φd is equal to 1− α(a+b)α2+χ is less than 1 and
depends on the preference parameter χ. In response to a shock to total reservedemand, reserve supply adjusts to fully accommodate the shift if φd = 1; thiswould succeed in insulating the funds rate from the shock, but it would lead totalreserves to move one-for-one with vd. By setting φd < 1, reserve supply lessthan fully accommodates the shift in reserve demand. This means the funds raterises (falls) if vd > 0 (< 0). This means the funds rate moves more, but totalreserves move less, and this will be optimal since the policy maker cares aboutboth E[TR]2 and E[if ]2
96
8 Chapter 10: Interest Rates andMonetary Pol-
icy
1. Suppose (10.1) is replaced by a Taylor sticky price adjustment model of thetype studied in Chapter 5. Is the price level still indeterminate under thepolicy rule (10.5)? What if prices adjust according to the Fuhrer-Mooresticky inflation model?
Equation (10.1) relates the level of real output to the price surprise termpt − Et−1pt; the actual level of prices doesn’t really matter. With a Taylor-typeprice adjustment model of the type discussed in section 5.5.1, the price level attime t will depend on prices in previous periods. Employing a simple versionin which prices are a markup over wages, and nominal wages are set for twoperiods with half of all wages set each period, the aggregate price level in periodt will be equal to
pt =1
2(xt + xt−1) (137)
where xt is the contract wage set in period t. If wage setting depends on theexpected price level over the two periods the wage is set and on the current stateof economic activity,
xt =1
2(pt +Etpt+1) + kyt (138)
as in equation (5.44) on page 216. Substituting (138) into (137),
pt =1
4(pt +Etpt+1 + pt−1 +Et−1pt) +
1
2k (yt + yt−1)
Multiplying both sides by 4 and rearranging, the Taylor adjustment model im-plies
pt =1
3(pt−1 +Etpt+1 +Et−1pt) +
2
3k (yt + yt−1)
If we combine this with the IS equation (10.7) and Fisher equation (10.8), thenunder an interest rate peg, equilibrium is obtained as the solution to
pt =1
3(pt−1 +Etpt+1 +Et−1pt) +
2
3k (yt + yt−1) (139)
yt = α0 − α1rt + ut (140)
iT = rt + (Etpt+1 − pt) (141)
To see if the price level is determinate, consider what would happen if, at the endof period t−1, the public expected the price level in all future periods to be higher
97
by κ%. Since the model is specified in log form, we need to check whether addingκ to pt, Et−1pt and Etpt+1 would affect the equilibrium. Clearly, equations(140) and (141) would be unaffected, (140) because it does not involve the pricelevel, and (141) because expected inflation is also unaffected if the price leveljumps by κ% and remains at this new higher level: (Etpt+1 + κ− (pt + κ)) =Etpt+1 − pt. But equation (139) is affected; pt−1 is predetermined — it can’tjump when expectations change. So when pt, Etpt+1 and Et−1pt increase byκ, the left side of 139) goes up be κ, while the right side only goes up by 2
3κbecause pt−1 can’t increase by κ; is no longer satisfied if pt jumps to pt + κ.Equilibrium is determined by the historical price level.
If a Fuhrer-Moore model of inflation adjustment is used, then equations (140)and (141) remain unchanged, but the inflation process is different. From equa-tion (5.61), the change in the contract wage is given by
∆xt =1
2(πt +Etπt+1) + 2kyt
and inflation is equal to
πt =1
2(∆xt +∆xt−1)
=1
4(πt +Etπt+1 + πt−1 +Et−1πt) + k (yt + yt−1)
Rearranging,
πt =1
3[πt−1 +Etπt+1 +Et−1πt] +
4
3k (yt + yt−1)
In terms of pt, this can be written as
pt = pt−1 +1
3[πt−1 +Etπt+1 +Et−1πt] +
4
3k (yt + yt−1) (142)
Again. history pins down the price level. A jump in pt and all future expectedprice levels leaves Etπt+1 and Et−1πt unaffected. Lagged inflation πt−1 is alsounaffected since it is predetermined as of time t. But pt also depends on the levelof past prices pt−1 so a κ% jump would not leave the equilibrium unaffected.
Both the Taylor and the Fuhrer-Moore models imply that the current pricelevel depends, in part, on the previous price level. Thus, the price level is deter-minate if we can take the historical value of pt−1 as given.
2. Derive the values of the unknown coefficients in (10.10) and (10.11) if themoney supply process is given by (10.15).
Equations (10.10) and (10.11) were used to derive the equilibrium processesfollowed by the price level and the nominal interest rate when the nominal money
98
supply was set according to (10.9). Suppose instead that mt is determined by(10.15), repeated here as
mt = µ′ + µ0t+ µ(it − iT
)The rest of model is given by equations (10.1) - (10.4). Using this new processfor mt, the proposed solutions for pt and it will need to be modified to allowfor the possibility that either the price level or the nominal rate of interest, orboth, may be affected by the deterministic trend that appears in the money supplyprocess. Thus, we need to consider the solutions
pt = b10 + b11mt−1 + b12et + b13ut + b14vt + b15t (143)
it = b20 + b21mt−1 + b22et + b23ut + b24vt + b25t (144)
Combining (10.1), (10.2), and (10.4) yields (10.12) for the nominal rate ofinterest:
it =α0 − yc
α1− 1
α1[a (pt − Et−1pt) + ut − et] + Etpt+1 − pt (145)
while (10.1), (10.3), and the new policy rule (10.15) yield
pt = µ′ + µ0t+ µ(it − iT
)+ cit − yc − a (pt − Et−1pt)− et − vt
= µ′ − µiT − yc + µ0t+ (µ+ c)it − a (pt − Et−1pt)− et − vt (146)
which can be compared to (10.13), obtained using the policy (10.9) in which themoney supply depended on lagged money.
We need to solve for the unknown coefficients in (143) and (144) such thatequations (145) and (146) are satisfied for all realizations of the e, u, and vand all t.
Using the proposed solutions,
pt − Et−1pt = b12et + b13ut + b14vt
and
Etpt+1 = b10 + b11mt + b15(t+ 1)
= b10 + b11µ′ + b15 − µiT + (b11µ0 + b15) t+ µit + b15(t+ 1)
Using these expressions for the expectational terms, together with the proposedsolutions, we can determine the unknown coefficients. For example, considerthe coefficients b11and b21 on mt−1 From (145), these must satisfy
b21 = −b11while from (146), they must satisfy
b11 = (µ+ c)b21
99
or
b11 = b21 = 0
Proceeding in a similar manner, the coefficients b12 and b22 on et mustsatisfy
b22 = − a
α1b12 − b12 − 1
α1
and
b12 = (µ+ c)b22 − ab12 − 1
or
b12 = −(
α1 + µ+ c
α1(1 + a) + (µ+ c)(α1 + a)
)
b22 =
(α1 − 1
α1(1 + a) + (µ+ c)(α1 + a)
)
For the coefficients on ut:
b23 = − a
α1b13 − b13 +
1
α1
and
b13 = (µ+ c)b23 − ab13
or
b13 =
(µ+ c
α1(1 + a) + (µ+ c)(α1 + a)
)
b23 =
(1 + a
α1(1 + a) + (µ+ c)(α1 + a)
)
For the coefficients on vt:
b24 = − a
α1b14 − b14
and
b14 = (µ+ c)b24 − ab14 − 1
or
b14 = −(
α1
α1(1 + a) + (µ+ c)(α1 + a)
)
100
b24 = −(
α1 + a
α1(1 + a) + (µ+ c)(α1 + a)
)
For the trend,
b25 = b15 − b15 = 0
and
b15 = µ0 + (µ+ c)b25 = µ0
Finally, for the constants
b20 =
(α0 − yc
α1
)+ b10 + b15 − b15
=
(α0 − yc
α1
)+ b10
b10 = µ′ + µiT +(µ+ c) (α0 − yc)
α1+ (µ+ c)µ0 − yc
Collecting these results, the equilibrium processes for the nominal interestrate and the price level are (ignoring the constant terms),
it = b20 +(α1 − 1) et + (1 + a)ut + (a+ α1)vt
α1(1 + a) + (µ+ c)(α1 + a)
pt = b10 + µ0t+(µ+ c)ut − (α1 + µ+ c) et − α1vt
α1(1 + a) + (µ+ c)(α1 + a)
These can be compared to the solution coefficients reported on page 437 and theinterest rate solution given in equation (10.14).
3. Suppose the money supply process in section 10.4.2 is replaced with
mt = γmt−1 + φqt−1 + ξt
so that the policy maker is assumed to response with a lag to the realrate shock, with the parameter γ viewed as a policy choice. Thus, policyinvolves a choice of γ and φ, with the parameter φ capturing the systematicresponse of policy to real interest rate shocks. Show how the effect of qton the one and two period nominal interest rates depends on φ. Explainwhy the absolute value of the impact of qt on the spread between the longand short rates increase with φ.
101
This problem, and the following one, both use the model of section 10.4.2;they differ in terms of the process followed by the nominal stock of money. Itwill be convenient to solve the model for a more general specification of mt,allowing one to then obtain the solutions for Problems 3 and 4 (as well as theresults in the text) as special cases.
The model consists of the following four equations (see equations 10.29 -10.31 of the text):
Rt = qt (147)
Rt =1
2[it − Etπt+1 +Etit+1 − Etπt+2] (148)
mt − pt = −ait + vt (149)
and
mt = γmt−1 + φqt−1 + ξt − ρξt−1 (150)
where Rt is the two-real period interest rate, qt is an exogenous real rate shock,it is the one period nominal rate, π is the inflation rate, the third equation is amoney demand relationship, and the money supply process used in section 10.4.2(equation 10.32) has been replaced by the one specified in the question. Noticethat the case considered in the text had φ = ρ = 0, Problem 3 considers the casewith ρ = 0, and Problem 4 sets γ = 1, φ = 0 (and notice that to distinguishbetween the coefficient on lagged money and that on the lagged shock ( ξt−1) Ihave renamed the latter ρ for the purposes of deriving the general solution; inProblem 4, mt has a coefficient of 1 and the coefficient on ξt−1is called γ).
To answer this question, we need to solve for the one and two-period nominalrates, together with the interest rate spread,
st ≡ It − it =1
2(Etit+1 − it)
Using (149) to eliminate the one-period nominal rate from equation (148),the price level prices must satisfy
qt =1
2
[pt −mt + vt
a− (Etpt+1 − pt)
+Et
(pt+1 −mt+1 + vt+1
a
)− Et (pt+2 − pt+1)
]or
2aqt = pt −mt + vt + apt +Etpt+1 − Etmt+1 − aEtpt+2
Solving for pt,
(1 + a)pt = 2aqt +mt − vt − Etpt+1 +Etmt+1 + aEtpt+2
102
Now use (150) to obtain
Etmt+1 = γmt + φqt − ρξt
so that
(1 + a)pt = 2aqt + (1 + γ)mt − vt + φqt − ρξt − Etpt+1 + aEtpt+2 (151)
from which we can guess that the solution for pt is of the form
pt = b1mt + b2qt + b3vt + b4ξt
Using this to evaluate (151),
(1 + a) (b1mt + b2qt + b3vt + b4ξt) = 2aqt + (1 + γ)mt − b1 (γmt + φqt + ρξt)
+ab1γ (γmt + φqt − ρξt)− vt + φqt − ρξt
since Etpt+1 = b1Etmt+1 = b1 (γmt + φqt − ρξt) and Etpt+2 = b1Etmt+2 =γb1Etmt+1 = γb1 (γmt + φqt − ρξt). This equilibrium expression holds for allrealizations of mt and the random disturbances if4
b1 =1
1 + a(1− γ)
b2 =
(1
1 + a
)(2a+
aφ
1 + a(1− γ)
)
b3 = − 1
1 + a
and
b4 = −(
a
1 + a
)(ρ
1 + a(1− γ)
)or
pt =
(1
1 + a(1− γ)
)mt +
1
1 + a
[(2a+
aφ
1 + a(1− γ)
)qt − vt −
(aρ
1 + a(1− γ)
)ξt
](152)
which can be compared to equation (10.34) of the text. The case in the text isobtained by setting φ = ρ = 0.
Now that we have the solution for the price level, the one-period nominalinterest rate is, from (149),
it =1
a(pt −mt + vt)
= −(
1− γ
1 + a(1− γ)
)mt +
(1
1 + a
)(2 +
φ
1 + a(1− γ)
)qt
−(
1
1 + a
)(ρ
1 + a(1− γ)
)ξt +
(1
1 + a
)vt (153)
4 In deriving the solution for b1, the fact that 1− γ2 = (1 + γ)(1− γ) is used.
103
This implies that
Etit+1 = −(
1− γ
1 + a(1− γ)
)Etmt+1
= −(
1− γ
1 + a(1− γ)
)(γmt + φqt − ρξt)
The two period rate is
It =1
2[it +Etit+1]
=1
2
{−(
1− γ2
1 + a(1− γ)
)mt +
(1
1 + a
)vt
}
+1
2
{(1
1 + a
)[2 +
φ (γ − a(1− γ))
1 + a(1− γ)
]qt
}
−1
2
(1
1 + a
)[ρ (γ − a(1− γ))
1 + a(1− γ)
]ξt (154)
while the spread is
St =1
2(Etit+1 − it)
=1
2
{((1− γ)2
1 + a(1− γ)
)mt −
(1
1 + a
)vt +
(1
1 + a
)(2− γ + a(1− γ)
1 + a(1− γ)
)ρξt
−(
1
1 + a
)[(2 +
(2− γ + a(1− γ))φ
1 + a(1− γ)
)qt
]}(155)
For the specific money process assumed for Problem 3, ρ = 0. Thus, theterms involving ξt all become equal to 0. Interest rates depend on φ becausewhen φ differs from zero, agents will adjust their forecast of the future moneysupply once they observe qt. Suppose φ > 0; the money supply is increasedin response to a positive shock to the real rate of interest. Then a positive qrealization causes an upward revision in the future money supply and the futureprice level. This raises expected future inflation and so the one-period rate itrises more than in the φ = 0 case (see equation 153). With 0 < γ < 1,the expected future money supply returns only gradually to its baseline after apositive q shock. From (155), a positive q shock lowers the spread, but theabsolute value of the impact increases with φ. To understand why, consider thecase in which γ = 1 so that the effect of q on m is permanent. Observing q > 0raises Etmt+1 and Etπt+1, contributing to the rise in it. But since the moneysupply is now expected to remain at this higher level, Etπt+2 is unchanged. Thelong rate rises less than the short rate and the difference between the two islarger if qt has a large impact on mt+1 (i.e., if φ is large). When | γ |< 1, theinitial rise in mt is expected to gradually be reversed, so Etπt+1 will actuallyfall, increasing it relative to Etit+1
104
4. Suppose the money supply process in section 10.4.2 is replaced with
mt =mt−1 + ξt − γξt−1
Does it depend on γ? Does It? Explain.
We can used equations (153), (154), and (155) from Problem 3 to answerthis question, simply modifying the parameters to reflect the new money supplyprocess. In particular, the coefficient on lagged money is now equal to 1 (thiscoefficient was γ in Problem 3) and the coefficient on the lagged real rate shockis now zero (φ = 0). Finally, the coefficient on ξt−1,which was called in ρProblem 3, is renamed γ in the current Problem.
With these changes, the solutions for the one-period nominal rate and thetwo-period nominal rate become
it =
(1
1 + a
)(2qt + vt − γξt) (156)
It =1
2
(1
1 + a
)[2qt + vt − γξt] (157)
The one-period nominal rate does depend on γ; a positive realization of ξtincreases the period t money supply. If this increase were permanent, the ex-pected rate of inflation would not be affected as the current and expected futureprice levels would rise in proportion to the increase in mt. With γ nonzero, how-ever, some of the change in mt is offset in period t+ 1 (Etmt+1 = mt − γξt).If 0 < ξ < 1, for example, the money stock is expected to be lower in t+1 thanin period t. This reduces expected inflation and the nominal rate falls.
The two-period nominal rate is
It =1
2(it +Etit+1) =
1
2it
since Etit+1 = 0. The future one-period rate is unaffected (since the moneysupply remains constant at its t + 1 value — Etmt+2 = mt − γξt), so the two-period rate moves half as much as the one-period rate.
5. Show that equation (10.43) implies rt =1
1+D
∑∞i=0
(D
1+D
)iEt
(ift+i − πt+1+i
).
Equation (10.43) is repeated here:
rt −D [Etrt+1 − rt] = ift − Etπt+1
which can be written as
rt =
(1
1 +D
)(ift − Etπt+1
)+
(D
1 +D
)Etrt+1
105
Updating this one period and using the result to eliminate Etrt+1 yields
rt =
(1
1 +D
)(ift − Etπt+1
)
+
(D
1 +D
)[(1
1 +D
)(Eti
ft+1 − Etπt+2
)+
(D
1 +D
)Etrt+2
]
Continuing to recursively substitute forward results in
rt =1
1 +D
∞∑i=0
(D
1 +D
)i
Et
(ift+i − πt+1+i
)
under the assumption that limi→∞(
D1+D
)iEtrt+i = 0.
6. Ball (1997) uses the following two equation model:
yt+1 = a1yt − a2rt + ut+1
πt+1 = πt + γyt + ηt+1
The disturbances ut and ηt are taken to be serially uncorrelated. At time t,the policy maker chooses rt, and the state variable at time t is πt+γyt ≡ κt.Assume the policy maker’s loss function is given by equation (10.54). Theoptimal policy rule takes the form θt = Aκt where θt ≡ a1yt−a2rt. Derivethe optimal value of A.
Note: In the text, the coefficient on rt in the definition of θt is incorrectlylabelled as a3.
First rewrite the model in terms of κt and θt:
yt+1 = θt + ut+1
πt+1 = κt + ηt+1
Though its choice of rt, the policy maker can determine θt, so it simplifies theproblem to simply treat θt as the policy instrument.
The loss function (10.54) can now be written as
L =1
2Et
∞∑i=1
βi[λ (θt + ut+1)
2 +(κt + ηt+1
)2]
The objective is to minimize this subject to the constraint that
κt+1 = πt+1 + γyt+1
= κt + γθt + ηt+1 + γut+1
106
Define the value function
V (κt) = minθt
{1
2λEt (θt + ut+1)
2 +1
2Et
(κt + ηt+1
)2+ βEtV
(κt + γθt + ηt+1 + γut+1
)}
Then the first order conditions include
λθt + γβEtV′ (κt + γθt + ηt+1 + γut+1
)= 0 (158)
V ′(κt) = κt + βEtV′ (κt + γθt + ηt+1 + γut+1
)(159)
Multiplying the second of these by γ and adding it to the first yields
λθt + γV ′(κt) = γκt
or
V ′(κt) = κt −(λ
γ
)θt
This implies
EtV′(κt+1) = Etκt+1 −
(λ
γ
)Etθt+1
= κt + γθt −(λ
γ
)Etθt+1
Substituting this back into (158),
λθt + γβ
[κt + γθt −
(λ
γ
)Etθt+1
]= 0
or
θt = −(
γβ
λ+ γ2β
)κt +
(βλ
λ+ γ2β
)Etθt+1
When policy is set at time t, κt summaries the state, so optimal policy, giventhe linear-quadratic structure, will be of the form θt = Aκt. Using this proposedpolicy θt = Aκt, and recalling that Etθt+1 = AEtκt+1 = A(1 + γA)κt, thisbecomes
Aκt = −(
γβ
λ+ γ2β
)κt +
(βλ
λ+ γ2β
)A(1 + γA)κt
which yields the following quadratic equation for A:
γβλA2 − (λ− βλ+ γ2β)A+ γβ = 0
107
the solutions of which are
A1 =
(λ− βλ+ γ2β
)+√(λ− βλ+ γ2β)2 + 4γ2β2λ
2γβλ
and
A2 =
(λ− βλ+ γ2β
)−√(λ− βλ+ γ2β)2 + 4γ2β2λ
2γβλ
To determine which of these solutions we want, note that
κt+1 = κt + γθt = (1 + γA)κt
so that κt will be stable only if A < 0 so that the coefficient 1+ γA is less than1. Now consider the product of the two solutions A1 and A2:
A1A2 =
(λ− βλ+ γ2β
)2 − [(λ− βλ+ γ2β)2
+ 4γ2β2λ]
(2γβλ)2
=−4γ2β2λ
4γ2β2λ2= − 1
λ< 0
so one solution must be positive, the other negative. We are looking for thenegative solution, which is A2, so our optimal policy rule is
θt =
(λ− βλ+ γ2β
)−√(λ− βλ+ γ2β)2 + 4γ2β2λ
2γβλ
κt
In terms of the interest rate actually set by the policy maker, we can use thedefinition of θt as a1yt − a2rt and κt as πt + γyt to obtain
rt =a1yt − θt
a2
=
(a1a2
)yt −
(1
a2
)A2 (πt + γyt)
which is in the form of a Taylor rule:
rt =
(a1 − γA2
a2
)yt −
(A2
a2
)πt
108
9 Typos
The most up-to-date list of known typos can be found through my web page athttp://econ.ucsc.edu/~walshc/ or by going directly to http://econ.ucsc.edu/~walshc/typos.html.
Chapter 8
1. Page 382, Problem 1: The coefficient on l∗ should be α, not a.
2. Page 384, Problem 8: The central bank’s loss function should be thediscounted sum of the single period loss function given by equation (8.2).
3. Page 384, Problem 11: The weights on output and inflation in the lossfunction should be (λ− θ) and (1+ θ) so that they sum to 1 regardless ofthe realization of θ. Also set a = 1.
Chapter 9
1. Page 429, Problem 4: The term involving output in the loss functionshould be the output gap, y−yn, rather than simply y. Alternatively, onecould just set yn = 0 as a normalization.
Chapter 10
1. Page 452, unnumbered equation at bottom of page: The expression for theimpact of a money supply shock on the long term nominal interest rateshould be multiplied by 1/2 (see the coefficient on m in equation 10.36).
2. Page 476, Problem 6. In the last line of the problem, the coefficient on rin the definition of θ should be a2, not a3 as appears.
109