程序代写代做代考 Lecture 9: Review

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: 1􏰄1􏰅 c =e−ρEt c (1+rt+1) t t+1 • Re-write Euler Equation as:1 1􏰄􏰄1􏰅 􏰂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 max􏰈u(Ct) s.t. 􏰈Ct ≤A0+􏰈Yt t=1 t=1 t=1 • Optimal for C1 = C2 = ··· = CT, the B.C. implies 1􏰂T􏰃 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: 1􏰂T􏰃 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=1yn+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