Lecture 9: Review
Patrick Macnamara
ECON60111: Macroeconomic Analysis University of Manchester
Fall 2020
Review
• I will talk about the structure of the final exam.
• I will review 8 topics – I will NOT cover all the material for the exam
1. Solow Model
2. Ramsey-Cass-Koopmans Model 3. Real Business Cycle Model
4. Consumption
5. Unemployment Theories
6. New Keynesian Model
7. Monetary Policy
8. Fiscal Policy
2/25
Final Exam
• Exam will consist of two parts (Section A and Section B). • Section A: Answer 1 of 2 questions
• Section B: Answer 2 of 3 questions.
• Final exam counts for 70% of your final mark.
• Exam will cover all material from this course.
• Mock exam available on Blackboard – solutions will be posted.
• See Blackboard for full details, including how to submit.
3/25
Topics 1-4
4/25
Solow Model
• Key equation
k ̇(t)=sf (k(t))−(n+g+δ)k(t)
• Steady state-state level of capital per effective worker: k∗: sf(k∗) = (n+g+δ)k∗
actual investment break-even investment
• If k ̸= k∗, tendency of economy to converge back towards k∗.
5/25
Solow Model: Dynamics
• Key assumptions: (1) n+g+δ>0
(2) f(0)=0,f′ >0,f′′ <0 (3) Inada conditions
6/25
Ramsey Model
• Two key equations: • Euler equation:
c ̇(t) = f′(k(t))−ρ−θg c(t) θ
• Law of motion for k:
(Remember, we’re assuming δ = 0)
• These two equations, describe how (c,k) evolve over time, given k(0) and c(0)
• k(0) pinned down by assumption
• c(0) determined endogenously – you can determine it by
figuring out which value of c(0) will converge along the saddle path to the steady state
k ̇(t) = f (k(t)) − c(t) − (n + g)k(t)
7/25
Ramsey Model: Saddle Path
A
B
D
8/25
RBC: Intratemporal Optimization
• RBC: general equilibrium model, built up from micro foundations with competitive markets.
• FOC:
marginal increase in utility marginal disutility
uc(ct,lt)wt = −ul(ct,lt)
from additional consumption from working more
• Assumingu(ct,lt)=lnct +bln(1−lt),substituteforuc
and ul:
wt=b 1 ct 1−lt
• Income and substitution effects:
• Substitution effect: given ct , ∆wt > 0 ⇒ ∆lt > 0
• Incomeeffect: givenwt,∆ct >0⇒∆lt <0
• Income and substitution effects cancel when ∆wt = ∆ct
9/25
RBC: Euler Equation
• FOC:
marg. dec. in utility from marg. inc. in utility from
lower cons. today higher cons. tomorrow
• Plug in expression for uc to get Euler Equation:
11 c =e−ρEt c (1+rt+1)
t t+1 • Re-write Euler Equation as:1
11 1 c =e−ρ Et c Et[1+rt+1]+Covt c ,1+rt+1
t t+1 t+1
For some intuition: assume Cov (1/ct+1, 1 + rt+1) = 0
⇒ When Etrt+1 ↑, household lowers ct, increases ct+1 1 Use that E (XY ) = (EX )(EY ) + Cov(X , Y ).
uc(ct,lt)=e−ρEt [uc(ct+1,lt+1)(1+rt+1)]
10/25
Consumption: PIH with No Uncertainty
• Household’s problem (assume u′ > 0, u′′ < 0):
TTT maxu(Ct) s.t. Ct ≤A0+Yt
t=1 t=1 t=1 • Optimal for C1 = C2 = ··· = CT, the B.C. implies
1T Ct=T A0+Ys ∀t
s=1
• Consumption determined by permanent income; time pattern of income not relevant for consumption decision
11/25
Consumption: PIH with Uncertainty
• Now assume uncertainty over Ys: T
s=t
• Euler Equation:
u′(Ct ) = Et [u′(Ct+1)]
• Assuming quadratic utility, u(C) = C − (a/2)C2, then
C1 = E1C2 = E1C3 = · · · = E1CT . Budget constraints imply:
1T C1=T A0+E1Yt
t=1
Consumption exhibits certainty equivalence: uncertainty has no effect on consumption/saving behavior
maxE u(C) s.t.A+C=A +Yfors=t,...,T
t
s s s s−1 s
12/25
Topics 5-8
13/25
Unemployment: Mortensen-Pissarides
• Key friction is that it is difficult to match workers to firms
• Economy consists of workers and jobs.
• Workers can be either employed (E) or unemployed (U). • Jobs can be either filled (F) or vacant (V).
• Worker’s utility is
w if employed and b if unemployed. Worker’s discount rate is r
• Profits per unit time from a job is
y −w −c if filled and −c if vacant
• Vacant jobs can be created freely
• Jobs end at rate λ (exogenous)
14/25
Unemployment: Mortensen-Pissarides
• Matches are CRS function of U and V : M=M(U,V)=kU1−γVγ, k>0,0<γ<1
• Job-finding rate:
• Vacancy-filling rate:
a = M(U,V) U
α = M(U,V) V
• After meeting, wage is determined via Nash bargaining • Free entry condition: VV (E ) = 0
15/25
Unemployment: Equilibrium
Equilibrium employment determined where rVV = 0. As E increases, a increases, α decreases and w increases.
16/25
New Keynesian Model: Labor Supply
• FOC for labor supply:
V′(Lt) = U′(Ct)Wt
Pt • AssumingU(C)=C1−θ,V(L)=BLγ
t
t 1−θ t γt
BLγ−1 =C−θWt t tPt
• UsingthatYt =Lt andCt =Yt:
Real wages positively related to output.
Wt =BYθ+γ−1 Pt t
(1)
17/25
New Keynesian Model: Euler Equation
• The Euler equation is
U′(Ct) = β(1 + rt)U′(Ct+1)
• Substitute for U′(·):
C−θ =β(1+r)C−θ.
t tt+1 • Taking logs and dividing by θ give
lnCt =lnCt+1−1ln[β(1+rt)] θ
• UsingthatCt =Yt,ln(1+rt)≈rt,wegetthe new Keynesian IS curve,
, a ≡ − ln(β)/θ (2) This implies an inverse relationship between rt and Yt.
lnYt =a+lnYt+1−1rt θ
18/25
New Keynesian Phillips Curve
• The new Keynesian Phillips curve.
where
κ= α [1−β(1−α)]φ 1−α
πt =κyt +βEtπt+1
(3)
• It is derived by aggregating the behavior of price-setting firms facing barriers to price adjustment.
• Higher output (via higher marginal costs) raises inflation, as does expected future inflation.
• Why does expected future inflation appear in this equation? It captures the effect of expected future marginal costs, which is relevant for the firm’s price setting decision today.
19/25
Monetary Policy in NK Model
• NK IS curve and NK Phillips curve:
y =E [y ]−1(i −E[π ])+uIS
t tt+1 θt tt+1 t π t = β E t [ π t + 1 ] + κ ( y t − y tn )
• Central bank (CB) would like to set it to stabilize departures of output from ytn and inflation from zero.
• If CB sets it = rtn, it can achieve both objectives, yt = ytn and πt = 0 for all t. This is the divine coincidence.
• However – when CB sets it = rtn, model is prone to sunspot equilibria – i.e., equilibria with self-fulfilling beliefs.
• Define y ̃t = yt − ytn; consider interest rate rule of the form i t = r t n + φ π E t [ π t + 1 ] + φ y E t [ y ̃ t + 1 ]
Multiple equilibria can be ruled out if CB promises to respond to deviations of expected inflation/output
20/25
Monetary Policy: Output and Inflation
• The canonical New Keynesian model exhibits no long run trade-off in achieving the inflation and output objectives of the policy-maker.
• Most CBs perceive a trade-off between stabilizing inflation and stabilizing the gap between output and desired output.
• What might cause the divine coincidence to fail?
• It turns out our assumption that CB targets yn instead of y∗
was crucial.
If CB targets y∗ > yn, CB willing to trade-off higher inflation to get output closer to y∗
21/25
Monetary Policy: Inflation Bias
• Consider model due to Kydland and Prescott (1977) and Barro and Gordon (1983)
• General problem faced by a policy-maker: 2
minL=1yn+b(π−πe)−y∗ +1a(π−π∗)2 π 2 2
y
Policy-maker controls π, which affects output through Lucas supply curve.
• Two ways π and πe can be determined:
(1) Rules: Policy-maker makes binding commitment about π
before πe is determined.
Equilibrium is π = π∗ and y = yn.
(2) Discretion: Policy-maker chooses π taking πe as given.
Equilibrium is π > π∗ and y = yn.
22/25
Fiscal Policy: Government B.C.
• Evolution of government debt:
D ̇ (t) = [G(t) − T(t)]+r(t)D(t)
primary deficit
• The government’s budget constraint:
∞ −R(t) ∞
(4)
−R(t)
e G(t)dt ≤ −D (0) + 00
e
T(t)dt (5) PDV of gov’t purchases PDV of taxes
This is the government’s B.C.; it incorporates debt choices
the government must make given the path of G(t) and T(t). • Re-write (5):
∞ −R(t) e
0
[T(t)−G(t)]dt ≥D(0)
Government must run primary surpluses large enough (in present value) to offset its initial debt.
23/25
Fiscal Policy: Ricardian Equivalence
• Use government B.C., (5), to re-write household’s B.C.:2
∞ −R(t) ∞ e C(t)dt ≤K(0)+D(0)+
00 ∞ −R(t) ∞ −R(t)
−R(t)
[W(t)−T(t)]dt
∞ −R(t) 000
e C (t )dt ≤ K (0) + e
Household’s B.C. depends on present value of G(t).
G (t )dt • Law of motion for total household assets, A ≡ K + D:
A ̇ = rA + W − T − C r[K+D]+W−T−C=K ̇ +[G−T]+rD
• Using A ̇ = K ̇ + D ̇ :
e
W (t )dt − e
A ̇ D ̇ ⇒ K ̇ = rK + W − G − C
G, not T, affects capital accumulation.
2We assume number of households H = 1 and total population L(t) = 1. 24/25
Fiscal Policy: Tax Smoothing
• Government problem, without uncertainty:
∞ 1 Tt min Yf
∞ Tt−Gt
s.t. =D
0
Choose taxes to minimize present value of distortion costs
t=0 (1+r)t t Yt
t=0 (1+r)t
subject to overall budget constraint.
• Government sets taxes so that:
Tt Tt+1 Tt f′=f′⇒=
Tt+1 Yt+1
Yt Yt+1 Yt
where f ′ (T /Y ) is the marginal distortion cost.
Tax rate is constant over time.
• Implication for primary deficit, as a percentage of GDP:
Gt−Tt =Gt −τ, whereτ=T0/Y0=T1/Y1=··· Yt Yt
Primary deficit when Gt/Yt is high
25/25