The document covers advanced macroeconomic concepts such as the IS-LM model, optimal money holdings, and fiscal policy stability. Detailed mathematical derivations are presented for various economic conditions, emphasizing the effects of parameters on output and real money demand. Additionally, it discusses models of government taxation and implications of debt, precautionary saving, and the Barro tax-smoothing approach.

1. Homework 5
1)
a) the IS curve: ln Yt= ln Y(t+1) – (1/Ɵ)rt
so the slope is: drt/dyt (is) = -Ɵ/Yt. That means that an increase
in Ɵ will result in a steeper curve.
LM curve: Mt/Pt = Yt^(Ɵ/v) (1+rt / rt)^(1/v)
Ln(Mt/Pt) = (Ɵ/v) ln Yt +(1/v)ln(1+rt) – (1/v)ln rt.
0 = (Ɵ/v)(1/Yt)dYt + (1/v)(1/(1+rt)) drt – (1/v)(1/rt)drt.
The slope is: drt/dyt (LM) = (Ɵrt(1+rt))/Yt. That means that an
increase in Ɵ will result in a steeper curve.
b) the curve IS is not affected by the value of V. while curve
LM shifts upwards, since a decrease in v will result in an
increase for the demand for real money.
c) IS is not affected byΓ(.)
optimal money holdings: BΓ’(Mt/Pt) = (it/(1+it)) U’(Ct)
B(Mt/Pt)^(-v) = (it/1+it) Yt^-Ɵ
Mt/Pt= B^(1/v) Yt^(Ɵ/v) (1+rt/rt)^(1/v)
So this means that the LM curve will shift downwards.
2)
a) AC= (PC/)+(αYP/2)i
AC/ = -(PC/^2) + (αYP/2)I = 0
C/^2 = αYi/2
So *=(2C/αYi)^(1/2)
b) average real money holdings:M/P= αY/2
M/P = (αY/2) (2C/αYi)^(1/2)
M/P= (αCY/2i)^(1/2)
Ln(m/p) = (1/2)(lnα+lnY+lnC-ln2-lni)
(1/(M/P))((M/P)/i) = -(1/2)(1/i)
Elasticity of real money with respect to i: ((M/P)/)(i/(M/P)) = -
1/2
The elasticity with respect to Y : ((M/P)/Y)(Y/(M/P)) = ½
Average real money holdings increase in Y, and decrease in i.

2. 4)
a)when p is at a level that generates maximum output, LS meets
LD.
b) when p is above the level that generates maximum output,
will cause unemployment.
7)
a)
b)i)
ii)
iii)
13)
a) the asset has an expected rate of return r. capital gain/loss
plus dividends per unit time = rvp. There is no dividends per
unit time while searching for the palm tree, and there is b
probability per unit time of capital gain of (vc-vp)-c. the
difference in the price of the asset is(vc-vp) and –c is what the
asset pays, so at the end we have rvp=b(vc-vp-c)
b) there is probability aL that a person will find another person
with a coconut and trade with that person and gain u̅ . the
difference in the price of the asset is (vp-vc). So we end up with
rvp=al(vp-vc+u̅ ).
c) vp=(rvc/aL)+vc-u̅ .
r((rvc/aL )+vc-u̅ )= b(vc-(rvc/aL)-vc+u̅ -c)
vc(r(r+aL+b))/aL = u̅ (r+b)-bc
the value of being in state C: vc= (aL(u̅ (r+b)-bc)) / r(r+aL+b)

3. the value of being in state p: vp= ((u̅ (r+b)-bc)/(r+aL+b)) +
(aL(u̅ (r+b)-bc)/r(r+aL+b)) - u̅
so finally
vc-vp = (bc+u̅ aL)/(r+aL+b).
e) vc-vp ≥c
vc-vp = (bc+u̅ a(b/a))/(r+a(b/a)+b) = (bc+bu̅ )/(r+2b)
(bc+bu̅ )/(r+2b) ≥ c
That means that
Bc+bu̅ ≥c and c(r+2b-b) ≤ bu̅
So finally we have
c≤ bu̅ / (r+b).
f) it is a steady-state equilibrium for no one who finds a tree to
climb it for any value of c>0.
Yes there are values of c which there is more than one steady-
state equilibrium for 0<c< bu̅ /(r+b)
Yes, L = b/a has a higher welfare than L=0. When L=0 people
don’t gain any utility since they don’t climb a tree and don’t
have a chance to trade with other people and gain a coconut.
0 1 2 3 4 5 -3 -2.2000000000000002 -
1.8 -1.8 -2.2000000000000002 -3
0 1 2 3 4 5 7 6.5 5.5 3.5 1
0 1 2 3 4 -2 -2.5 -3.5 -5.5 -8
LD 1 2 3 4 5 6 7 8 9 10 10 9
8 7 6 5 4 3 2 1 LS 1 2 3

4. 4 5 6 7 8 9 10 1 2 3 4 5
6 7 8 9 10
AD 2 3 4 5 4 3 2 1 1 2 3 4
5 6 7 8 9
0 1 2 3 4 5 6 7 8 9 10 9 8
7.5 7 6.7 6.5 6.3 6.1 5.9
0 1 2 3 4 5 -2.6 -2 -1.4 -
1.1000000000000001 -0.95 -0.87 0 1
0 1 2 3 4 7 5 4.3 3.5 2
Advanced macroeconomics, 4th edition. Romer.
Chapter12.
12.1. The stability of fiscal policy. (Blinder and Solow, 1973.)
By definition, the budget deficit equals the rate of change of the
amount of debt outstanding: δ(t) ≡ D ̇(t). Define d(t) to be the
ratio of debt to output: d(t) = D(t)/Y(t). Assume that Y(t) grows
at a constant rate g > 0.
(a) Suppose that the deficit-to-output ratio is constant: δ(t)/Y(t)
= a, where a > 0.
̇
(i) Find an expression for d(t) in terms of a, g, and d(t). ̇
(ii) Sketch d(t) as a function of d(t). Is this system stable?
(b) Suppose that the ratio of the primary deficit to output is
constant and equal to a > 0. Thus the total deficit at t, δ(t), is
given by δ(t) = aY(t) + r(t)D(t), where r(t) is the interest rate at

5. t. Assume that r is an increasing function of the debt-to-output
ratio: r(t) = r(d(t)), where r′(•) > 0, r′′(•) > 0, limd→−∞ r(d) <
g, limd→∞ r(d) > g.
̇
(i) Find an expression for d(t) in terms of a, g, and d(t). ̇
(ii) Sketch d(t) as a function of d(t). In the case where a is
sufficiently small that d ̇ is negative for some values of d, what
are the stability properties of the system? What about the case
where a is sufficiently large that d ̇ is positive for all values of
d ?
12.2. Precautionary saving, non-lump-sum taxation, and
Ricardian equivalence.
(Leland, 1968, and Barsky, Mankiw, and Zeldes, 1986.)
Consider an individual who lives for two periods. The
individual has no initial wealth and earns labor incomes of
amounts Y1 and Y2 in the two periods. Y1 is known, but Y2 is
random; assume for simplicity that E[Y2] = Y1. The
government taxes income at rate τ1 in period 1 and τ2 in period
2. The individual can borrow and lend at a fixed interest rate,
which for simplicity is assumed to be zero. Thus second-period
consumption is C2 = (1 − τ1)Y1 − C1 + (1 − τ2)Y2. The
individualchoosesC1
tomaximizeexpectedlifetimeutility,U(C1)+E[U(C2)].
(a) Find the first-order condition for C1.
(b) Show that E[C2] = C1 if Y2 is not random or if utility is
quadratic.
(c) Show that if U ′′′(•) > 0 and Y2 is random, E[C2] > C1.
(d) Suppose that the government marginally lowers τ1 and
raises τ2 by the same amount, so that its expected total revenue,
τ1Y1 + τ2E[Y2], is un- changed. Implicitly differentiate the
first-order condition in part (a) to find an expression for how
C1 responds to this change.
(e) Show that C1 is unaffected by this change if Y2 is not
random or if utility is quadratic.
(f) Show that C1 increases in response to this change if U ′′′(•)

6. > 0 and Y2 is random.
12.3
Consider the Barro tax-smoothing model. Suppose that output,
Y, and the real interest rate, r, are constant, and that the level of
government debt out- standing at time 0 is zero. Suppose that
there will be a temporary war from time 0 to time τ. Thus G(t)
equals GH for 0 ≤ t ≤ τ, and equals GL there- after,whereGH
>GL.Whatarethepathsoftaxes,T(t),andgovernmentdebt
outstanding, D(t)?
12.4
Consider the Barro tax-smoothing model. Suppose there are two
possible val- ues of G(t)—GH and GL—with GH > GL.
Transitions between the two values follow Poisson processes
(see Section 7.4). Specifically, if G equals GH, the probability
per unit time that purchases fall to GL is a; if G equals GL, the
probability per unit time that purchases rise to GH is b. Suppose
also that output, Y, and the real interest rate, r, are constant and
that distortion costs are quadratic.
(a) Derive expressions for taxes at a given time as a function of
whether G equals G H or G L , the amount of
debt outstanding, and the exogenous parameters.
(Hint: Use dynamic programming, described in Section 10.4, to
find an expression for the expected present value of the revenue
the government must raise as a function of G, the amount of
debt outstanding, and the exogenous parameters.)
(b) Discuss your results. What is the path of taxes during an
interval when G equals GH? Why are taxes not constant during
such an interval? What happens to taxes at a moment when G
falls to GL? What is the path of taxes during an interval when G
equals GL?
(Romer 640)
12.9.
Consider the Tabellini–Alesina model in the case where α can

7. only take on the values 0 and 1. Suppose, however, that there
are 3 periods. The period-1 median voter sets policy in periods
1 and 2, but in period 3 a new median voter sets policy. Assume
that the period-1 median voter’s α is 1, and that the probability
that the period-3 median voter’s α is 1 is π.
(a) Does M1 = M2?
(b ) Suppose that after choosing purchases in period 1, the
period-1 median voter learns that the probability that the
period-3 median voter’s α will be1isnotπ butπ′,whereπ′ <
π.Howdoesthisnewsaffecthisor her choice of purchases in period
2?
12.10. ThePersson-
Svenssonmodel.(PerssonandSvensson,1989.)Suppose there are
two periods. Government policy will be controlled by different
policy- makers in the two periods. The objective function of the
period-t policymaker is U + αt [V(G1) + V (G2)], where U is
citizens’ utility from their private consumption; αt is the weight
that the period-t policymaker puts on public consumption; Gt is
public consumption in period t; and V(•) satis-
fiesV′(•)>0,V′′(•)<0.Privateutility,U,isgivenbyU
=W−C(T1)−C(T2), where W is the endowment; Tt is taxes in
period t; and C(•), the cost of raising revenue, satisfies C′(•) ≥
1, C′′(•) > 0. All government debt must be
paidoffattheendofperiod2.ThisimpliesT2
=G2+D,whereD=G1−T1 is the amount of government debt
issued in period 1 and where the interest rate is assumed to
equal zero.
(a) Find the first-order condition for the period-2 policymaker’s
choice of G2 given D. (Note: Throughout, assume that the
solutions to the policy- makers’ maximization problems are
interior.)
(b) How does a change in D affect G2?
(c) Think of the period-1 policymaker as choosing G1 and D.
Find the first-
order condition for his or her choice of D.

8. (d) Show that if α1 is less than α2, the equilibrium involves
inefficiently low taxation in period 1 relative to tax-smoothing
(that is, that it has T1 < T2). Explain intuitively why this
occurs.
(e) Does the result in part (d) imply that if α1 is less than α2,
the period-1 policymaker necessarily runs a deficit? Explain.
Chapter 6.
6.15. Observational equivalence. (Sargent, 1976.) Suppose that
the money supply isdeterminedbymt =c′zt−1
+et,wherecandzarevectorsandet isani.i.d. disturbance
uncorrelated with zt−1. et is unpredictable and unobservable.
Thus the expected component of m t is c ′ zt −1 , and the
unexpected component is et. In setting the money supply, the
Federal Reserve responds only to vari- ables that matter for real
activity; that is, the variables in z directly affect y .
Now consider the following two models: (i ) Only unexpected
money mat- ters,so yt = a′zt−1+bet+vt;(ii)allmoneymatters,so yt
= α′zt−1+βmt +νt.In each specification, the disturbance is i.i.d.
and uncorrelated with zt −1 and et.
(a) Is it possible to distinguish between these two theories? That
is, given a candidate set of parameter values under, say, model
(i), are there param- eter values under model (ii ) that have the
same predictions? Explain.
(b) Suppose that the Federal Reserve also responds to some
variables that donotdirectlyaffectoutput;thatis,supposemt
=c′zt−1+γ′wt−1+et and that models (i) and (ii) are as before
(with their distubances now uncorrelated with wt −1 as well as
with zt −1 and et). In this case, is it pos- sible to distinguish
between the two theories? Explain.
6.16. Consider an economy consisting of some firms with
flexible prices and some with rigid prices. Let pf denote the

9. price set by a representative flexible-price firm and pr the price
set by a representative rigid-price firm. Flexible-price firms set
their prices after m is known; rigid-price firms set their prices
be- fore m is known. Thus flexible-price firms set pf = pi∗ =
(1 − φ)p + φm, and rigid-price firms set pr = Epi∗ = (1 − φ)Ep
+ φEm, where E denotes the expectation of a variable as of
when the rigid-price firms set their prices.
Assume that fraction q of firms have rigid prices, so that p =
qpr+ (1−q)pf.
(a) (b) (c)
Find pf in terms of pr,m, and the parameters of the model (φ
and q). Find pr in terms of Em and the parameters of the model.
(i ) Do anticipated changes in m (that is, changes that are
expected as of when rigid-price firms set their prices) affect y ?
Why or why not?
(ii ) Do unanticipated changes in m affect y ? Why or why not?