# 程序代写代做代考 chain matlab Lecture 3:

Lecture 3:

Real Business Cycle Model

Patrick Macnamara

ECON60111: Macroeconomic Analysis University of Manchester

Fall 2020

Some Business Cycle Facts

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Business Cycle Facts (Part 1)

• Fluctuations are irregular in frequency and magnitude

• Time between end of one recession and beginning of next

ranges from 4 quarters (in 1980-1981) to about 10 years

(1960-1970, 1991-2000)

• Fall in Real GDP in 2001 very small (if anything),

but 4% fall during 2007-09, and likely to be very large during Covid-19 recession

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US Real GDP Growth

15.0

10.0

5.0

0.0

-5.0

-10.0

-15.0

1950 1960 1970

1980 1990 2000 2010 2020 Year

NBER recessions highlighted. Source: BEA NIPA

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Annual Real GDP Growth (%)

US Real GDP Per Person

60,000 50,000

40,000

30,000

20,000

1950 1960 1970

1980 1990 Year

2000 2010 2020

NBER recessions highlighted. Source: BEA NIPA

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Real GDP per person (chained 2012 dollars, log scale)

Business Cycle Facts (Part 2)

• Fluctuations are distributed very unevenly over the components of output.

• e.g., consumption, investment, etc. • Components fluctuating a lot:

• inventory investment

• durable consumption

• residential investment (i.e., housing)

• fixed non-residential investment (i.e., business investment)

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Components of GDP

Component of GDP Consumption

Durables Nondurables Services

Investment

Residential

Fixed nonresidential Inventories

Average share in GDP

63.3%

8.5% 19.5% 35.3%

17.2%

4.7% 12.0% 0.5%

Average share in fall

in GDP in recessions relative to normal growth

35.6%

15.0% 9.5% 11.1%

78.7%

11.0% 22.0% 45.7%

-0.8% -13.5%

Government purchases 20.7%

Net exports

-1.2%

Source: Romer textbook, Table 5.2

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Real GDP Growth vs. Consumption

15.0

10.0

5.0

0.0

-5.0

-10.0

-15.0

Y C

1950 1960 1970

1980 1990 Year

2000 2010 2020

Correlation = 0.80, consumption less volatile. Source: BEA NIPA

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Annual Growth (%)

Real GDP Growth vs. Investment

80.0

60.0

40.0

20.0

0.0

-20.0

-40.0

Y I

1950 1960 1970

1980 1990 2000 Year

2010 2020

Correlation = 0.81, investment more volatile. Source: BEA NIPA

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Annual Growth (%)

Real GDP Growth vs. Government Spending

50.0

40.0

30.0

20.0

10.0

0.0

-10.0

Y G

1950 1960 1970

1980 1990 2000 Year

2010 2020

Correlation = 0.21. Source: BEA NIPA

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Annual Growth (%)

Real GDP Growth vs. Net Exports

15.0

10.0

5.0

0.0

-5.0

-10.0

Y NX

1950 1960 1970

1980 1990 Year

2000 2010 2020

Correlation = 0.02. Source: BEA NIPA

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Annual Growth (%)

Business Cycle Facts (Part 3)

• No large asymmetries between rises and falls in output; Distribution of output growth (roughly) symmetric

• Most of the time, output is growing; actual declines in Real GDP relatively infrequent

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Histogram of Real GDP Growth

50

40

30

20

10

0

-10 -5 0 5 10

Annual Real GDP Growth (%)

Source: BEA NIPA

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Count

Business Cycle Facts (Part 4)

Avg. change in recessions -4.2%

-2.5%

+1.9 p.p. -2.8%

-1.6% -0.2 p.p. -0.4% -1.8 p.p. -1.5 p.p. -0.1%

5.3

1Cyclicality of labor productivity

mid-1980s. See Table 1 in Fernald and Wang (2016).

Variable

Real GDP Employment Unemployment rate Average weekly hours

production workers, manuf. Output per hour1

Inflation

Real wages

Nom. interest rate Real interest rate Real money stock

Source: Romer textbook, Table

Number of recessions in which var. falls 11/11

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0/11 11/11

10/11

4/11

7/11

10/11

10/11

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(output per hour) has declined since

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Model Assumptions

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RBC Model

• Real Business Cycle (RBC) theory due to Kydland and Prescott (1982), Long and Plosser (1983), Prescott (1986)

• RBC model is a general equilibrium model, built up from microeconomic foundations with competitive markets

• This differs from Keynesian models at the time (i.e., IS-LM)

• Important contribution is methodological

(see Lucas critique, Lucas 1976)

• Shocks are real, not monetary

• No market imperfections

• Later work builds on the RBC model

• Adding market frictions

• New Keynesian models follow the same approach:

build model with micro foundations, add sticky prices

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Main Ingredients of the Model

(1) Discrete time

(2) Two types of optimizing agents:

• Firms

• Households • Government

(3) Several competitive markets:

• Goods market: all-purpose output good • Labor market

• Rental market for capital goods

(4) Two shocks:

• Shock to aggregate productivity • Shock to government spending

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Firms

• Representative firms, take prices as given, hire labor and rent capital to maximize profits in competitive markets

• Assume Cobb-Douglas production function:

Yt =Ktα(AtLt)1−α, 0<α<1
Inputs: capital (K), labor (L) and technology (A)
• Firms maximize profits:
Π =maxKα(AL)1−α−wL −(r +δ)K tKt,Ltttt tttt
• FOC imply
AtLt 1−α
rt=αK −δ t
Kt α
wt=(1−α) AL At tt
• Profits Πt = 0 in equilibrium
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Firms: Technology
• Technology grows at rate g, but subject to random shocks lnAt =A ̄+gt+A ̃t
A ̃t = ρAA ̃t−1 + εA,t εA,t ∼i.i.d. N(0,σA2)
• Assume−1<ρA <1
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Households
• H identical households
• Total population is Nt, which grows at exogenous rate n:
lnNt=N ̄+nt, Nt=eN ̄+nt
• Each household has Nt/H members
• Households own the representative firms
• Households own capital, rent it out to firms
• Household, taking prices as given, choose consumption, labor supply and saving to maximize lifetime utility.
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Households: Utility
• Household’s utility function:
∞ Nt
U=Ee−ρtu(c,l) , ρ>n 0 ttH

t=0 Definition of terms:

• ρ = discount rate, e−ρ = discount factor

• ct = consumption of each household member • lt = labor supply of each household member • u(ct,lt) = instantaneous utility function

• Assume u(ct,lt) given by

u(ct,lt)=lnct +bln(1−lt), b>0

• Aggregate consumption Ct and aggregate labor Lt: Ct=ct×Nt×H ⇒ ct=Ct/Nt

H

Lt=lt×Nt×H ⇒ lt=Lt/Nt H

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Households: Budget Constraint

• Capital evolves according to

Kt+1 = (1 − δ)Kt + It, δ = rate of depreciation

• Budget constraint

Ct +Kt+1 −(1−δ)Kt =wtLt +(rt +δ)Kt +Πt −Tt

It

where Tt = Gt is a lump-sum tax levied by the government (balanced budget).2

• Divide through by H, use Ct = ctNt and Lt = ltNt:

c Nt +Kt+1 =wl Nt +(1+r)Kt +Πt −Tt tHHttHtHHH

2UsingTt =Gt,Yt =wtLt +(rt +δ)Kt andΠt =0,thebudget constraint constraint implies that Yt = Ct + It + Gt .

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Households: Maximization Problem

• Household solves (given prices, K0, Nt, Πt, Tt)

∞ Nt

max E e−ρtu(c ,l ) s.t.

{ct,lt,Kt+1}∞ 0 t t H t=0 t=0

c Nt +Kt+1 =wl Nt +(1+r)Kt +Πt −Tt tHHttHtHHH

• Budget constraint holds for all t and for all potential realizations for the shocks (A ̃t,G ̃t) in the future.

• Household has rational expectations: they make the best possible prediction about future wages wt, interest rates rt, and shocks (A ̃t,G ̃t).

• Solution to this problem is not just one sequence {ct,lt,Kt+1}∞t=0. Solution is a contingent plan: household makes choices for (ct , lt , Kt+1) at every t, which are contingent on the whole history of shocks up to t!

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Government

• Government levies lump-sum tax, runs a balanced budget Tt = Gt

• Government purchases grow at rate n + g , and are subject to random shocks:

lnGt =G ̄+(n+g)t+G ̃t G ̃ t = ρ G G ̃ t − 1 + ε G , t

εG,t ∼i.i.d. N(0,σG2) • Assume−1<ρG <1
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Rational Expectations Equilibrium
• The rational expectations equilibrium is represented by prices{wt,rt}∞t=0 andallocations{ct,lt,Kt+1}∞t=0 suchthat
(1) Firms optimize, given prices
i.e., allocations satisfy the firm’s maximization problem
(2) Households optimize, given prices
i.e., allocations satisfy the household’s maximization problem
(3) Markets clear
• Capital supply equals capital demand • Labor supply equals labor demand
(4) Household’s forecasts of future prices and shocks coincide with their actual law of motion
• Since first welfare theorem applies, competitive equilibrium is Pareto efficient
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Analyzing the Model
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Household’s Problem: Lagrangian
• Using Πt = 0, Lagrangian for household’s problem:3
+
∞ Nt KtKt+1Tt E e−ρtλ (wl −c) +(1+r) − −
∞ Nt L = E e−ρt u(c , l )
0 ttH t=0
0ttttHtHHH t=0
• FOC:
∂L =[uc(c0,l0)−λ0]N0 =0 ∂c0 H
∂L =[ul(c0,l0)+λ0w0]N0 =0 ∂l0 H
∂L λ0 (1+r1) ∂K =−H+E0 e−ρλ1 H
1
=0
3Multiplying λt by e−ρt has no effect on the solution and only matters for
how we interpret λt.
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Household’s Problem: FOC
• FOC (from previous slide) ul(c0,l0)+λ0w0 =0
λ0 =uc(c0,l0)
λ0 = e−ρE0 [λ1(1 + r1)]
What about more generally, for any t? • Two ways of looking at the problem:4
• At date 0, household chooses (ct,lt,Kt+1) for all t and for every possible realization of the shocks (A ̃t,G ̃t).
• At date 0, household chooses (c0, l0, K1) given (A ̃0, G ̃0), K0. At date 1, after learning (A ̃1, G ̃1), choose (c1, l1, K2),
And so on...
• Re-write the household’s problem, starting from date t
• Problem at date t looks just like the problem at date 0 4Another way to look at the problem is to use dynamic programming.
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Household’s Problem: Lagrangian Re-Written
• Household’s Lagrangian, starting from date t:
∞ Nt+s
L=E e−ρs[u(c ,l )+λ (w l −c )]
t t+st+s t+st+st+st+sH
s=0
∞ Kt+s Kt+s+1
Tt+s t t+st+sHHH
+Ee−ρsλ(1+r)− − s=0
• FOC:
∂L =[uc(ct,lt)−λt]Nt =0 ∂ct H
∂L =[ul(ct,lt)+λtwt]Nt =0 ∂lt H
∂L λt (1+rt+1) ∂K =−H +Et e−ρλt+1 H
t+1
(1)
(2) =0 (3)
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Optimal Labor Supply
• From (1) and (2):
marginal increase in utility marginal disutility
uc(ct,lt)wt = −ul(ct,lt)
from additional consumption from working more
• Plug in expressions for uc 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
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Optimal Labor Supply: Further Intuition
• Optimal tradeoff between consumption and labor supply: wt=b 1
ct 1−lt
• Temporary decrease in technology (i.e., fall in A ̃t): • Wage wt falls (hence wages are procyclical)
• Since decrease in technology is temporary,
fall in ct < fall in wt (i.e., households smooth consumption)
⇒ labor supply lt falls
• Suggests reductions of labor supply in recessions are optimal
responses to lower productivity
• Permanent increases in technology (i.e., trend growth): • wt and ct increase by same amount
⇒ no effect on labor supply
(income and substitution effects cancel)
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Euler Equation
• From (1) and (3):
uc(ct,lt)=e−ρEt [uc(ct+1,lt+1)(1+rt+1)]
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:5
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
When Cov (1/ct+1, 1 + rt+1) < 0, saving is less attractive 5 Use that E (XY ) = (EX )(EY ) + Cov(X , Y ).
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Solving the Model
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Log-Linearization
• In general RBC models can only be solved numerically, like many other modern models in macroeconomics.
• A common way of solving RBC (and other) models is to log-linearize the model:
• Approximate model equations (e.g., decision rules and laws of motion for state variables) with first-order Taylor approximation around the non-stochastic steady state
• A lot of tools available that make this easy • Uhlig’s Matlab toolkit – link
• Schmitt-Grohe and Uribe’s Matlab toolbox – link • Dynare – link
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Log-Linearization: General Procedure
(1) Find equations characterizing equilibrium (e.g., constraints, FOCs, etc.)
(2) Pick parameters and find non-stochastic steady state
(3) Construct log-linear approximations of the equations
characterizing the equilibrium
(4) Solve for the recursive equilibrium law of motion using
the method of undetermined coefficients.
(5) Analyze the solution.
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Determine Non-Stochastic Steady State
• Model with no shocks converges to balanced growth path. Let y∗, k∗, c∗, w∗, r∗, l∗ denote the constant values of
Y/(AL), K/(AL), C/(AL), w/A, r, l constant
• Define “tilde” variables as the log deviation from the
non-stochastic steady state. For example:
K ̃t =lnKt −lnKt∗ L ̃t =lnLt −lnL∗t
I use stars to denote the balanced growth path variable (which may not be constant).
• Note that, for each variable X:
X ̃t = ln Xt − ln Xt∗ ≈ Xt − Xt∗ Xt∗
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Log-Linearizing Model Equations
• Log-linearized resource constraint and FOCs:6
K ̃ −λK ̃ −λ A ̃ −L ̃−λG ̃ −λC ̃ =0 (4)
t+1 1t 2 t t 3t 4t ̃ ̃ ̃ ̃l∗ ̃
α Kt −Lt +(1−α)At −Ct −1−l∗Lt =0 (5) EλA ̃ +L ̃ −K ̃ +C ̃−C ̃ =0 (6)
t R t+1 t+1 t+1 t t+1 where (λ1, λ2, λ3, λ4, λR ) are constants that depend on
model parameters.
• Assumed structure of shocks:
A ̃t+1 = ρAA ̃t + εA,t+1; Et[εA,t+1] = 0 (7) G ̃t+1 = ρGG ̃t + εG,t+1; Et[εG,t+1] = 0 (8)
6I leave it as an exercise for students to derive these equations.
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Solving the Log-Linearized Model
• We have system of equations with two types of variables: • 3 state variables: K ̃t, A ̃t, G ̃t
• 2 control variables: C ̃t, L ̃t
• Guess that solution takes the following form:
C ̃t =aCKK ̃t +aCAA ̃t +aCGG ̃t L ̃t =aLKK ̃t +aLAA ̃t +aLGG ̃t
K ̃t+1 =bKKK ̃t +bKAA ̃t +bKGG ̃t
• Method of undetermined coefficients: solve for (a, b) such that the model equations ((4) to (8)) are satisfied.
• You don’t need do this by hand – e.g., use Matlab tools
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Implications
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Calibration: Why?
• Kydland and Prescott (1982), Long and Plosser (1983) illustrate the value of exploring models using a “reasonable” set of parameter values.
• Following recommendation of Lucas (1980), Kydland and Prescott relied on micro empirical studies and long-run properties of the economy to choose parameter values.
• Once we fix the parameter values, we can
• investigate the transmission mechanism of the shocks. • see whether the model matches the data
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Calibration In Practice
• We assume that each period is a quarter.
• Baseline parameter values (Campbell, 1994):7
α = 1/3
g = 0.5% n = 0.25%
δ = 2.5% ρA = 0.95 ρG = 0.95
• Set G ̄, ρ, and b are such that
(G/Y)∗ = 0.2, r∗ = 1.5%, and l∗ = 1/3.
7We don’t need to set σA or σG when computing impulse responses.
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Effect of a 1% Technology Shock
1
0.8
0.6
0.4
0.2
0
Technology Shock: Capital and Labor
A K
L
-0.2
0 10 20 30 40 50 60
Quarters
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Pct. Dev. from Steady State
Effect of a 1% Technology Shock
1
0.8
0.6
0.4
0.2
0
Technology Shock: Consumption and Output
C Y
-0.2
0 10 20 30 40 50 60
Quarters
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Pct. Dev. from Steady State
Effect of a 1% Technology Shock
1
0.8
0.6
0.4
0.2
0
Technology Shock: Wage and Interest Rate
w r
-0.2
0 10 20 30 40 50 60
Quarters
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Pct. Dev. from SS (w) / Pct. Pt. Diff. from SS (r)
Effect of a 1% Government-Purchases Shock
0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2
Government Shock: Capital and Labor
K
L
0 10 20 30 40 50 60 Quarters
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Pct. Dev. from Steady State
Effect of a 1% Government-Purchases Shock
0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2
Government Shock: Consumption and Output
C
Y
0 10 20 30 40 50 60 Quarters
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Pct. Dev. from Steady State
Effect of a 1% Government-Purchases Shock
0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2
Government Shock: Wage and Interest Rate
w
r
0 10 20 30 40 50 60 Quarters
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Pct. Dev. from SS (w) / Pct. Pt. Diff. from SS (r)
RBC Model vs. Data
• Consider Hansen and Wright (1992) calibration • Similar parameters as before
Moment U.S. data
σY 1.92
RBC model
1.30 0.31 3.15 0.49 0.93
σC /σY
σI /σY
σL /σY Corr(L, Y /L)
0.45 2.78 0.96
-0.14
All variables are de-trended using HP filter. Source: Hansen and Wright (1992)
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Summary of Main Implications
Prescott (1986):
• “Economic theory implies that, given the nature of the shocks to technology and people’s willingness and ability to intertemporally and intratemporally substitute, the economy will display fluctuations like those the U.S. economy displays.”
• “[T]heory predicts what is observed. Indeed, if the economy did not display the business cycle phenomena, there would be a puzzle.”
• “Economic fluctuations are optimal responses to uncertainty in the rate of technological change.”
• “The policy implication of this research is that costly efforts at stabilization are likely to be counterproductive.”
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Criticism
• Strong evidence that monetary shocks have real effects
• Friedman and Schwartz (1963), Romer and Romer (1989)
• RBC model omits channel through which other disturbances
have real effects
• Technology shocks are the dominant source of fluctuations
• Solow residual poor measure of technology shocks.8
• Empirical evidence that properly identified technology
shocks cause labor to fall.9
• To perform well, the model requires an empirically unreasonably high intertemporal elasticity of substitution for labor supply
• For example, see Martinez Saez Siegenthaler (2020)
8However, see King and Rebelo (1999).
9For example, see Francis and Ramey (2005), Basu, Fernald and Kimball (2006), Chang and Hong (2006) and Chang, Hornstein, Sarte (2009).
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