RBC Model - Universitetet i oslo · RBC Model I Analyzes to what extend growth and business cycles...
Transcript of RBC Model - Universitetet i oslo · RBC Model I Analyzes to what extend growth and business cycles...
RBC Model
I Analyzes to what extend growth and business cycles can begenerated within the same framework
I Uses stochastic neoclassical growth model (Brock-Mirmanmodel) as a workhorse, which is augmented by a labor-leisurechoice
I Business cycles reflect optimal response to stochasticmovements in the evolution of technological progress. No rolefor monetary factors in explaining fluctuations (’real’ businesscycles)
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What are we going to do?
I We will document several empirical regularities (”stylizedfacts”) of business cycles
I We will use the standard neoclassical growth model as a toolto understand the causes of business cycles
I Using a model as a measurement tool requires 3 steps:I Mapping the model’s parameters to the data: Calibration
I Solving the model (we will use log-linearization)
I Comparing the model’s outcome and the stylized facts
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Real GDP in U.S.I Want to understand aggregate economic activity: real GDP
Figure: Kruger, Quantitative Macroeconomics: An Introduction4 / 96
Use of Logarithms
I Assume variable Y grows at a constant rate g
I It follows that Yt = (1 + g)tY0
I Taking (natural) logarithms
log(Yt) = log [(1 + g)tY0]
= log(Y0) + log [(1 + g)t ]
= log(Y0) + t ∗ log [1 + g ]
I If Y grows at a constant rate g , it will be a straight line withslope log [1 + g ] ≈ g for small g
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Use of Logarithms: Taylor Expansion
I The fact that log(1 + g) ≈ g is the result of a Taylor seriesexpansion of log(1 + g) around g = 0:
log(1 + g) = log(1) +g − 0
1− 1
2(g − 0)2 +
1
6(g − 0)3 + . . .
= g − 1
2g 2 +
1
6g 3 + . . .
≈ g
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Isolating Cycles & Removing Trends
I Business cycles = deviations from long-run growth trend
I Let Yt be real GDP. Then
log(Yt) = log(Y trend) + log(Y cycle)
I We are interested in the cyclical component:
log(Y cycle) = log(Yt)− log(Y trend)
I How to detrend the data?
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Isolating Cycles & Removing Trends
I Different filters that perform this task
I Detrending
I First-difference filter
I Hodrick-Prescott (HP) filter
I And others (Bandpass, . . . )
I They differ with respect to assumptions about the trendcomponent
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Removing Trends: Detrending
I Assume that trend is deterministic:
Yt = (1 + g)tY0eut , ut ∼ (mean-zero, stationary)
I Taking log’s (using log(1 + g) ≈ g)
log(Yt) = log(Y0) + gt + ut
I The cyclical component log(Y cycle) is given by
log(Y cycle) = ut = log(Yt)− log(Y0)− gt
I log(Y0) and g can be estimated by OLS
I Deterministic trend assumption has been challenged in thetime-series literature (see e.g. Nelson/Plosser 1982)
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Removing Trends: Differencing
I Assume that trend is stochastic:
Yt = Y0eεt
εt = g + εt−1 + ut , ut ∼ (mean-zero, stationary)
I Iterative substitution for εt−1, εt−2, . . . yields
εt = gt +t−1∑j=0
ut−j + ε0
I The cyclical component log(Y cycle) is given by
log(Y cycle) = ut = log(Yt)− log(Yt−1)− g
I This can be achieved by taking first differences & demeaningthe sample average of log(Yt)− log(Yt−1).
I This implicitly assumes constant average growth rate g
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Removing Trends: HP-Filter
Solve the following minimization problem:
minlog(Y trend
t )
T∑t=1
(log(Yt)− log(Y trendt ))2
+λT∑t=1
[(log(Y trendt+1 )− log(Y trend
t ))− (log(Y trendt )− log(Y trend
t−1 ))]2
Results depend on λ. One can show that if
I λ = 0: log(Yt) = log(Y trendt )
I λ→∞ : log(Y trendt ) = log(Y trend
t−1 ) + g
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HP-Filtered Real GDP
Figure: λ = 1600,Kruger (2007). Quantitative Macroeconomics: AnIntroduction 12 / 96
Detrended GDP
Figure: Solid: Det. Trend, Dots: Diff’ed, Dashes: HP (DeJong/Dave(2007).Structural Macroe’metrics)
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Summary: Removing Trends & Isolating Cycles
I Cyclical component looks very different depending on ourassumptions
I Choice of filter somewhat arbitrary
I To evaluate model: eliminate trends from the data generatedby the model and actual data in the same way
I When working with quarterly data, be aware of seasonality.Adjust the data before filtering
I In the following: Look at HP-filtered data
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Stylized Facts
We are interested in
I the amplitude of fluctuations
I the degree of comovement with real GNP
I whether there is a phase shift of a variable relative to theoverall business cycle, as defined by cyclical real GNP
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Stylized Facts
Some labels:
I If the contemporaneous correlation coefficient of a variablewith real GNP is positive (negative), we say it is procyclical(countercyclical)
I A variable leads the cycle if correlation coefficient of the serieswhich is shifted forward w.r.t. real GNP is positive
I A variable lags the cycle if correlation coefficient of the serieswhich is shifted backward w.r.t. real GNP is positive
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Stylized Facts
Some observations:
I Fluctuations in consumption and capital are smoother thanoutput fluctuations
I Investment is much more volatile than output
I Total hours worked are almost as volatile as output
I The real wage and the real interest rate are quite smooth
I Consumption, investment and hours worked are veryprocyclical
I Productivity is also procyclical, but much less volatile thanoutput
I Wages are uncorrelated with output
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The Basic RBC Model: Introduction
I To what extend can stochastic neoclassical growth modelaccount for these facts?
I We ’discipline’ the model by making it consistent withlong-run growth
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The Basic RBC Model: Introduction
Model consists of
I Households
I Firms
I Other sectors (i.e. government) could be added
Recall Brock-Mirman economy we discussed in Macro I
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The Basic RBC Model: Introduction
Households (HH)
I A large number of identical, infinitely lived HH
I HH maximize utility which they derive from consumption ofgoods and consumption of leisure (or disutility of work)
I HH supply labor to firms and rent out capital to firms
I HH use their income either for consumption or for buyinginvestment goods which they add to their capital stock
I HH behave competitively taking all prices for given
There is a representative household
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The Basic RBC Model: Introduction
Firms
I A large number of identical firms
I Firms rent capital and labor from households
I They produce a single good and take all prices as given
I Assume that they operate a constant returns to scaletechnology
Perfect competition and constant returns to scale imply that thenumber of firms is indetermined: representative firm
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The Basic RBC Model
Representative HH problem:
maxct ,ht
E0
[ ∞∑t=0
βtu(ct , 1− ht)
]
such that
kt+1 + ct = wtht + (1 + rt)kt
0 ≤ ct
0 ≤ kt+1
k0given
Recall that we could use sequence formulation to make uncertaintymore explicit, as we did in Macro I.
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The Basic RBC Model
Remarks
I Rational expectations imply that household computesexpectations using the ’correct’ probabilities
I Notice that there are only aggregate shocks (that affect thewhole economy) but no idiosyncratic shocks (that affect theindividual households differently)
I During the course we will also study the opposite case (noaggregate but idiosyncratic shocks)
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The Basic RBC ModelWe can make use of the welfare theorems and study the planner’sproblem:
maxct ,ht
E0
[ ∞∑t=0
βtu(ct , 1− ht)Nt
]such that
Ct = ctNt
Kt+1 + Ct = ZtF (Kt ,AtNtht) + (1− δ)Kt
At+1 = (1 + gA)At
Nt+1 = (1 + gN)Nt
Zt+1 = Z ρt eεt , ρ ∈ (0, 1), εt ∼ N(0, σ2)
0 ≤ Ct
0 ≤ Kt+1
K0,Z0,N0,A0, given
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The Basic RBC Model: Existence of BalancedGrowth Path
I Balanced growth: growth in output, capital and consumption(per capita) grow over long periods of time
I Balanced growth is characteristic for most industrializedcountries
I Long-run growth occurs at rates that are roughly constantover time (but may differ across countries)
I We need to impose certain restrictions on functional forms toguarantee existence of balanced growth path
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The Basic RBC Model: Existence of BalancedGrowth Path
Where does economic growth come from?
I We think of increases in output at given levels of inputthrough increase in ’technological knowledge’ which we takeas exogenous
I Can be either labor-augmenting or capital augmenting
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The Basic RBC Model: Existence of BalancedGrowth Path
Technology:
I Impose labor-augmenting technological progress At and aproduction function that features constant returns to scale:
Yt = ZtF (Kt ,AtNtht)
where λYt = ZtF (λKt , λAtNtht)
I We will typically work with Cobb-Douglas technology
Yt = ZtKαt (AtNtht)
1−α
I Here, technical progress can always be written as purelylabor-augmenting
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The Basic RBC Model: Existence of BalancedGrowth Path
Some notation:
I Define growth factor of variable V
γV ≡Vt+1
Vt= 1 + gV
I Express variables in per-capita terms: yt ≡ YtNt
, kt ≡ KtNt
,
ct ≡ CtNt
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The Basic RBC Model: Existence of BalancedGrowth Path
From resource constraint:
γk =kt+1
kt=
yt − ct + (1− δ)kt(1 + gN)kt
I On balanced growth path, γk is constant
I This implies that ytkt
and ctkt
are constant as well
I Thus γk = γy = γc on balanced growth path
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The Basic RBC Model: Existence of BalancedGrowth Path
Verify existence on balanced growth path under the assumptionabout technology above:
I
γy =yt+1
yt= γk
F (1,Xt+1)
F (1,Xt)
where Xt ≡ Athtkt
I From this, we get γy = γkγF and γX = γAγhγk
I Therefore,γk = γy ⇒ γF = 1⇒ γX = 1
I Hence, γk = γAγh
I Notice that γh = 1 (otherwise h→ 1 which is inconsistentwith balanced growth)
I Therefore γk = γy = γc = γA on balanced growth path
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The Basic RBC Model: Existence of BalancedGrowth Path
I Return on labor supply:
w ≡ AtF2(k
A, h)
I Along the balanced growth path, γk = γA
I Therefore, γw = γA
I How can it be that w is growing but labor supply is constant(γh = 1)?
I Need to impose restrictions on preferences s.t. income andsubstitution effect of permanent increase in w cancel out
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The Basic RBC Model: Existence of BalancedGrowth Path
I Return on capital:
r ≡ F1(k
A, h)
which is constant along the balanced growth path
I Euler equation implies
u1(ct , 1− ht)
u1(ct+1, 1− ht+1)= β(1 + r − δ)
where the RHS is constant along the balanced growth path
I Since γc = γA, consumption grows at a constant rate
I It follows that marginal utility of consumption has to changeat a constant rate as well → intertemporal elasticity ofconsumption independent of c
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The Basic RBC Model: Implications of BalancedGrowth Path
The following utility function are consistent with a balancedgrowth path:
1.
u(ct , 1− ht) =(ct(1− ht)
θ)1−σ − 1
1− σwith θ, σ ≥ 0
2. and
u(ct , 1− ht) = log(ct) + θ(1− ht)
1+κ
1 + κ
with θ, κ ≥ 0
3. oru(ct , 1− ht) = log(ct) + θlog(1− ht)
with θ ≥ 0
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The Basic RBC Model: Implications of BalancedGrowth Path
These specifications yield to the following optimality conditions forthe intratemporal trade-off between consumption and leisure:
1.θct
1− ht= wt
2.θhκt ct = wt
3.θct
1− ht= wt
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The Basic RBC Model: Implications of BalancedGrowth Path
I Substitution effect: Increase in wt makes leisure moreexpensive
I Income effect: higher wages mean - for unchanged laborsupply - higher income
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The Basic RBC Model: Implications of BalancedGrowth Path
I Consider the budget constraint in a static world (nointertemporal effects): ct = htwt
I Plugging this into FOCs above, we find that effect of wt
cancels out
I Income and substitution effects cancel out
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The Basic RBC Model: Implications of BalancedGrowth Path
I Recall that on balanced growth path, increase in w arepermanent and r is constant
I Households budget constraint is the same as in the static case(see graph ”golden rule level of capital stock”)
I Income and substitution effects of wage changes cancel out
I No effect on labor supply
I Hence, γw = γA = γc and γh = 1
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The Basic RBC Model: Implications of BalancedGrowth Path
I Restriction on preferences has important implications for theability of the model to generate fluctuations
I If capital is absent or if wages grow permanently, there is noendogenous response to exogenous productivity (King, Plosserand Rebelo 1988)
I Intertemporal substitution, stemming from temporary changesin productivity and transmitted through capital are key forgenerating amplification in the RBC model
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The Basic RBC Model: Implications of BalancedGrowth Path
Given these restrictions, it is possible to define new variables thatare constant in the long-run:
kt =Kt
(1 + gA)t(1 + gN)t=
kt(1 + gA)t
yt =Yt
(1 + gA)t(1 + gN)t=
yt(1 + gA)t
ct =Ct
(1 + gA)t(1 + gN)t=
ct(1 + gA)t
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The Basic RBC Model: Stationary Version
maxct ,ht
E0
[ ∞∑t=0
βt(ct(1− ht)
θ)1−σ − 1
1− σNt
]such that
(1 + gA)(1 + gN)kt+1 + ct = Zt kαt h1−α
t + (1− δ)kt
βt = βt(1 + gA)t(1−σ)
Zt+1 = Z ρt eεt , ρ ∈ (0, 1), εt ∼ N(0, σ2)
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The Basic RBC Model: First-Order Conditions
Euler-Equation:
1+gA = βEt
[(αZt+1kα−1
t+1 h1−αt+1 + 1− δ
)( ctct+1
)σ (1− ht+1
1− ht
)θ(1−σ)]
(1)Intra-temporal labor-leisure trade-off:
θct1− ht
= (1− α)Zt kαt h−αt (2)
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Calibration: Introduction
I We want to know to what extend the model replicatesbusiness cycle facts
I Select parameter values such that model can be used as ameasurement tool
I Select parameters such that deterministic version of model(no productivity shocks) is consistent with empirical factsabout long-run growth
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Calibration: Strategy
2 sets of parameter values
I Direct empirical counterpart: estimated from the data
I No direct empirical counterpart: calibrated to match long-runaverages in the data
Some remarks:
I Distinction not always clear-cut
I Often disagreement about the ’correct’ parameter value
I Robustness checks to assess sensitivity of results should thusbe good practice
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Calibration: Long-Run Growth Rates
I Growth rates measure the change from one period to the next
I Need to decide about the length of a period in model
I Business Cycle analysis usually done on quarterly data
I Population growth gN : ∼ 1.1% per year. Per Quarter:
gN = (1.011)14 − 1 ≈ 0.0027
I Growth of GDP per capita gA: ∼ 2.2% per year. Per Quarter:
gA = (1.022)14 − 1 ≈ 0.0055
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Calibration: Curvature Utility Function
I Determined by σ: Higher values imply higher degree of riskaversion & stronger incentive for smooth consumption profile
I Estimates are between σ = 1 & σ = 3
I σ = 1 common in business cycle literature
I This implies u(ct , 1− ht) = log(ct) + θ log(1− ht)
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Calibration: Technology yt = Zt kαt h
1−αt
I Long-run mean of Zt is Z ≡ 1
I α is given by the capital share in total output
sk ≡ rK
Y=
kαkα−1h1−α
y= α
I In the data, sk is constant over time and amounts to 30 - 40percent of total output
I Exact value depends on the treatment of income fromself-employment, of housing and the government sector
I Here: α = 0.4
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Calibration: Depreciation Rate δ
I On balanced growth path: kt = kt+1 = k
I Budget constraint:
(1 + gA)(1 + gN)kt+1 = (1− δ)kt + (yt − ct)︸ ︷︷ ︸(1 + gA)(1 + gN)k = (1− δ)k + i
δ =i
k+ 1− (1 + gA)(1 + gN)
I i
k= 0.076 on an annual level
δ =0.076
4+ 1− (1 + 0.0055)(1 + 0.0027) ≈ 0.012
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Calibration: Discount Factor β
I On balanced growth path: ct = ct+1 = c
I With σ = 1, β = β
I The Euler-Equation simplifies to
(1 + g) = β
(α
y
k+ 1− δ
)I k
y ≈ 3.32 on an annual level. The quarterly ratio is3.32 ∗ 4 = 13.28
I Using α, δ, g we get β = 0.987
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Calibration: Weight of Leisure θ
I Rewrite condition 2
(1− α)y
c= θ
h
1− h
I h = 0.31: households spend 13 of their time working
I yc = 1.33
I This yields θ = 1.78
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Approximation Methods
I Model is very complex - in general it is not possible to deriveexplicit solutions
I Need to rely on approximation techniques
I We will learn two approaches:I Make use of recursive structure and write down problem as a
dynamic programm. Use value function iteration toapproximate decision rules
I Directly work on the model’s optimality conditions. Problem:non-linearity. Solution: (Log-)linear approximation ofoptimality conditions
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Approximation Methods: Log-Linearization
I Here, we will log-linearize optimality conditions
I Approximate solution around steady-state
I Variables are expressed in % deviation from steady-state →unit-free!
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Log-Linearization
I Determine Constraints and FOCs
I Compute steady-state
I Log-linearize necessary conditions
I Solve for recursive equilibrium law of motion via the methodof undetermined coefficients
I Analyze the solution via impulse-response analysis andsimulation of second moments
This follows the Uhlig (1997) procedure closely (also seehomework).
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Example
Social Planner Problem:
maxCt
E0
[ ∞∑t=0
βtC
(1−σ)t
1− σ
]
such that
Kt+1 + Ct = ZtKαt + (1− δ)Kt
Zt+1 = Z ρt eεt , ρ ∈ (0, 1), εt ∼ N(0, σ2)
0 ≤ Ct
0 ≤ Kt+1
K0,Z0given
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Example: Necessary Conditions
Ct + Kt+1 = ZtKαt + (1− δ)Kt
Rt = αZtKα−1t + (1− δ)
C−σt = Et
[βC−σt+1Rt+1
]Zt+1 = Z ρ
t eεt , ρ ∈ (0, 1)
+ transversality condition
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Example: Steady State
Z = 1
C + K = Kα + (1− δ)K ⇔ C = Y − δK
R = αKα−1 + (1− δ)⇔ K =
(α
R − 1 + δ
) 11−α
1 = βR ⇔ R =1
β
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Log-Linearization
Necessary conditions can be re-written in terms of an implicitfunction:
f (x , y) = 0
where x and y are steady state values of x and y . By implicitdifferentiation
∂f (x , y)
∂xdx +
∂f (x , y)
∂ydy = 0
or
∂f (x , y)
∂xx
dx
x+∂f (x , y)
∂yy
dy
y= 0 (3)
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Log-Linearization
I dyy = y−y
y ' log(
1 + y−yy
)= log y
y ≡ y
I y : % deviation from steady-state
I Re-write (3):
x
[∂f (x , y)
∂xx
]+ y
[∂f (x , y)
∂yy
]' 0 (4)
I Linear in x and y
I Alternatively, take log’s first and then perform first-orderTaylor expansion around log(x) and log(y)
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Log-Linearization: Let’s Do It!
Budget Constraint:
Kt+1 + Ct − ZtKαt − (1− δ)Kt = 0
This is a function in 4 variables: Kt+1, Ct , Zt and Kt
Applying (4) gives
K kt+1 + C ct − Z Kα(zt + αkt)− (1− δ)K kt ≈ 0
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Log-Linearization: Let’s Do It!
Euler-Equation:
βEt
[C−σt+1Rt+1
]− C−σt = 0
Contains 3 variables: Ct+1, Rt+1 and Ct
Applying (4) and using 1 = βR yields
βEt
[−σC (−σ−1)C ct+1R + C−σR rt+1
]+ σC (−σ−1)C ct ≈ 0⇔
Et [−σct+1 + rt+1] + σct ≈ 0⇔Et [σ(ct − ct+1) + rt+1] ≈ 0
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Log-Linearization: Let’s Do It!
Return-Function:
Rt − αZtKα−1t − (1− ρ) = 0
3 variables: Rt , Zt and Kt
Applying (4) yields
R r − αZ Kα−1(zt + (α− 1)kt) ≈ 0
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Collecting Equations
1.K kt+1 + C ct − Z Kα(zt + αkt)− (1− δ)K kt = 0
2.Et [σ(ct − ct+1) + rt+1] = 0
3.R r − αZ Kα−1(zt + (α− 1)kt) = 0
4.zt+1 = ρzt + εt
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Log-Linearization
I Determine Constraints and FOCs X
I Compute steady-state X
I Log-linearize necessary conditions X
I Solve for recursive equilibrium law of motion via the methodof undetermined coefficients
I Analyze the solution via impulse-response analysis andsimulation of second moments
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Method of Undetermined Coefficients
I We want to find policy functions: recursive law of motion
I We have to solve system of linear differential equations, whichis given by the log-linearized equilibrium conditions
I Use Method of Undetermined Coefficients
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Method of Undetermined Coefficients
We postulate a linear recursive law of motion
kt+1 = νkk kt + νkz zt
rt = νrk kt + νrz zt
ct = νck kt + νcz zt
Solve for the ”undetermined” coefficients
νkk , νkz , νrk , νrz , νck , νcz
Similar approach to ”Guess and Verify”
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Method of Undetermined Coefficients
Let’s see how it works. The necessary condition for the interestrate is given by
R r − αZ Kα−1(zt + (α− 1)kt) = 0
which we can re-write to
r − (1− β(1− δ))(zt − (1− α)kt) = 0 (5)
by making use of1
β= R = αZ Kα−1
(5) depends on parameter values only
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Method of Undetermined Coefficients
We can now determine the coefficients of the policy function for rt :
r = (1− β(1− δ))(zt − (1− α)kt)
νrk kt + νrz zt = (1− β(1− δ))(zt − (1− α)kt)
νrk kt + νrz zt = (1− β(1− δ))zt − (1− β(1− δ))(1− α)kt
thus
νrk = −(1− β(1− δ))(1− α)
νrz = (1− β(1− δ))
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Method of Undetermined Coefficients
I Proceed similar manner for the other equations
I After a while, you’ll end up with a quadratic equation in νkk :
0 = ν2kk − γνkk +
1
β(6)
where
γ =(1− β(1− δ))(1− α)(1− β + βδ(1− α))
σαβ+ 1 +
1
β
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Method of Undetermined Coefficients
I Equation (6) has two solutions
I We are looking for |νkk | < 1: stable root
I If |νkk | > 1, k keeps growing (falling) which will violatetransversality condition (the non-negativity constraint)
I Use stable root to calculate
νkz , νrk , νrz , νck , νcz
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Log-Linearization
I Determine Constraints and FOCs X
I Compute steady-state X
I Log-linearize necessary conditions X
I Solve for recursive equilibrium law of motion via the methodof undetermined coefficients X
I Analyze the solution via impulse-response analysis andsimulation of second moments
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Log-Linear Approximation: Appraisal
I Works almost always → has become standard procedure inthe literature
I Computationally very fast, but linearization tedious
I Local method as optimal policies are computed nearsteady-state: works only for small deviations
I Implicitly assumes certainty equivalence
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Log-Linear Approximation: Certainty Equivalence
Log-linear version of Euler-Equation:
Et [σ(ct − ct+1) + rt+1] ≈ 0⇔σct ≈ Et [σct+1 + rt+1]
Compare this to the deterministic Euler equation:
σct ≈ σct+1 + rt+1
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Log-Linear Approximation: Certainty Equivalence
I The property that the decision rule depends only on the firstmoment of the distribution that characterize uncertainty iscalled certainty equivalence
I Higher moments (e.g. variance) do not matter for the choices
I This is a problem if ’true’ solution depends on highermoments (e.g. if there is precautionary saving)
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Alternative Methods
Alternative local solution methods:I Optimal Linear Regulator
I Excellent alternative for social planner problems, avoids tediouslinearization
I Second-order approximation (Schmitt-Groh/Uribe 2004)I Does not impose certainty equivalence
Global solution methods such as successive approximation of thevalue/policy function
I Compute optimal choice for all feasible values of the statevariables
I Precise but slow
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Recursive Law of Motion
I After this long detour, we return to our model withendogenous labor
I Using the calibrated parameters, we can compute the policyfunctions with the help of the procedure outlined before
kt = 0.97kt−1 + 0.08zt
ct = 0.63kt−1 + 0.31zt
ht = −0.27kt−1 + 0.81zt
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Recursive Law of Motion
I We can make use of this to trace out the response of oureconomy to technology shocks: ”Impulse responses”
I We can shock the economy repeatedly and trace out theresponses: ”Simulation”
I Useful for understanding the qualitative and quantitativeproperties
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Technology Shocks
Production Function
yt = Ztkαt (1 + g)th1−α
t
whereZt+1 = Z ρ
t eεt , ρ ∈ (0, 1), εt ∼ N(0, σ2)
We want to estimate ρ and σ2
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Technology Shocks
Taking logs
log(yt) = log(Zt) + αlog(kt) + (1− α)log(ht) + (1− α)tlog(1 + gA)
log(Zt) = log(yt)− (αlog(kt) + (1− α)log(ht) + (1− α)tlog(1 + gA))
I Zt is the ”Solow-Residual”
I Estimate ρ and σ from log(Zt) = ρlog(Zt−1) + εt
I In the data, techn. shocks are quite persistent: ρ = 0.95
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Impulse Responses
I In t = 0, set all variables to 0
I In t = 1, technology shock ε1 > 0
I In t = 2, ...,T , εt = 0. Trace out kt and zt using theirrecursive law of motion
I Given kt and zt for t = 2, ...,T , trace out all other variables
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Simulation
I Given σ Simulate sequence of ε′ts using a random numbergenerator
I Pick some initial k0 and z0
I Calculate recursively
zt+1 = ρzt + εt
kt+1 = νkk kt + νkz zt
I With that, obtain all other variables
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RBC Mechanism
I How does the economy react to a temporary increase inproductivity?
I Response of labor supply is particularly important: change inht determines whether the model amplifies or dampens thefluctuations generates by zt
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RBC Mechanism
I The FOC’s of the representative household in our case are:
θct1− ht
= wt (7)
where wt ≡ (1− α)Ztkαt h−αt and
I Euler-Equation:
1 = βEt
[Rt+1
(ct
ct+1
)](8)
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RBC Mechanism
I We can combine (7) and (8) to get
1− 1
βEt
1
Rt+1
wt+1
wt︸ ︷︷ ︸Wt
1− ht+1
1− ht
= 0 (9)
I where Wt is the wage growth in present value terms
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RBC Mechanism
I Recall that on a balanced growth path, wt grows at aconstant rate, hence wt+1
wtis constant
I Moreover, Rt+1 = R on a balanced growth path
I Hence, W = W
I Therefore 1−ht+1
1−ht is constant
I By construction, this has to hold for all utility functionsconsistent with balanced growth path !
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RBC Mechanism
In general, labor supply depends on
I the relative wage. If w1 is higher than w2 (because of atemporary productivity shock), households supply more labortoday than tomorrow
I the interest rate. A higher interest rate induces households toincrease their labor supply today as returns are higher
I The sensitivity of these effects depends on the intertemporalelasticity of substitution (which is 1 in this example)
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RBC Mechanism
I The IES of labor supply is given by
d 1−ht+1
1−htdWt
Wt
1−ht+1
1−ht
=dln(
1−ht+1
1−ht
)dln (Wt)
= 1
I If c and 1− h are additively separable in the utility function,the IES of labor supply is identical to the Frisch elasticity oflabor supply
I Estimates using micro data suggest that the Frisch elasticity isbelow 0.5
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Baseline
-1 0 1 2 3 4 5 6 7 8-1
0
1
2
3
4
5
6
7Impulse responses to a shock in technology
Years after shock
Per
cent
dev
iatio
n fro
m s
tead
y st
ate
capital consumption
output
labor
interest
investment
technology
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Low Peristence (ρ = .8)
-1 0 1 2 3 4 5 6 7 8-1
0
1
2
3
4
5
6
7
8Impulse responses to a shock in technology
Years after shock
Per
cent
dev
iatio
n fro
m s
tead
y st
ate
capital consumption
output
labor
interest
investment
technology
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Baseline
0 5 10 15 20 25 30 35 40-20
-15
-10
-5
0
5
10
15
20
capital consumption
output labor interest
investment
technology
Simulated data (HP-filtered)
Year
Per
cent
dev
iatio
n fro
m s
tead
y st
ate
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Baseline
-5 -4 -3 -2 -1 0 1 2 3 4 5 Std. Devoutput -0.03 0.03 0.12 0.36 0.67 1 0.67 0.36 0.12 0.03 -0.03 1.2519
capital -0.45 -0.41 -0.36 -0.23 -0.02 0.29 0.48 0.56 0.57 0.56 0.53 0.22
cons. -0.23 -0.17 -0.07 0.16 0.5 0.9 0.73 0.53 0.35 0.28 0.22 0.2891
labor 0.02 0.09 0.17 0.4 0.69 0.99 0.62 0.29 0.05 -0.04 -0.11 0.6904
interest 0.05 0.11 0.19 0.41 0.69 0.99 0.6 0.27 0.02 -0.07 -0.13 0.0316
investment 0.01 0.08 0.16 0.39 0.68 1 0.63 0.31 0.06 -0.02 -0.09 5.4904
techno. -0.01 0.05 0.14 0.37 0.67 1 0.65 0.33 0.1 0.01 -0.06 0.8422
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RBC Assessment
Kydland and Prescott (1982), Nobel Prize Laureates (2004):
”A competitive equilibrium model was developed and used toexplain the autocovariances of real output and the covariances ofcyclical real output with other aggregate economic timeseries...results indicate a surprisingly good fit in light of themodel’s simplicity”.
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RBC: Assessment
I Output fluctuates quite a bit, but less than in the data
I Consumption, investment and labor input are very procyclical,as in the data
I Investment is much more volatile, as in the data
I Factor prices are quite smooth, as in the data
I However, labor input is less volatile than output
I Correlation of all variables with output is very high, too highcompared to the data
I Productivity is nearly as volatile as output (low internalpropagation of model)
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RBC: Reasons for Model’s ’Weakness’
I Technology shock is very persistent, therefore wages adjustsmoothly, generating little fluctuations in labor
I As a result, too little fluctuations in labor input and weakinternal propagation
I It should be noted that the implied IES of labor by the modelis in stark contrast to the estimates from the micro data
I The high correlation of all variables with output is due to thefact that there is only one shock
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Extensions: Labor Markets
I Generating realistic fluctuations in aggregate labor supplywithout imposing an IES on the individual level is a bigchallenge
I See problem set for a solution that was proposed by Hansen(1985)
I Moreover, there is no notion of ’unemployment’ in thefrictionless RBC model
I Modeling unemployment can be an important mechanism togenerate amplification and persistence (see Hall (1998: LaborMarket Frictions and Employment Fluctuations)
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Extensions: TFP Shocks
I Are they correctly measured?
I What is their interpretation? (Are deep recessions reallyperiods of ’technical regress’?)
I Are technical shocks really exogenous with respect to policy?
I See King and Rebelo (1999): Resuscitating Real BusinessCycles and Rebelo (2005): Real Business Cycle Models: Past,Present, and Future
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Extensions: Asset Prices and FinancialIntermediation
I Counterfactual behavior of asset prices
I More recently: How to incorporate money and financialintermediation without imposing rather than explaining it (asthe New Keynesians do)?
I See Kiyotaki and Moore (2009): Liquidity, Business Cycles,and Monetary Policy and Gertler and Kiyotaki (2009):Financial Intermediation and Credit Policy in Business CycleAnalysis
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