RBC Model - Universitetet i oslo · RBC Model I Analyzes to what extend growth and business cycles can be generated within the same framework I Uses stochastic neoclassical growth

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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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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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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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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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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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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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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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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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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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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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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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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I Substitution effect: Increase in wt makes leisure moreexpensive

I Income effect: higher wages mean - for unchanged laborsupply - higher income

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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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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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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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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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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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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


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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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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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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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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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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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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


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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

88 / 96

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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Documents

RBC Model - Universitetet i oslo · RBC Model I Analyzes to what extend growth and business cycles can be generated within the same framework I Uses stochastic neoclassical growth