Phd Macro, 2007 (Karl Whelan) 1

Real Business Cycle Models

The Real Business Cycle (RBC) model—introduced in a famous 1982 paper by Finn Kyd-

land and Edward Prescott—is the original DSGE model.1 The early rational expectations

models of Lucas and Sargent had followed in the Keynesian tradition of making monetary

and fiscal policies a central feature of business cycle theory. However, those models were

generally not deemed to be empirically successful. RBC modelers took a different direction:

Their approach takes the microfoundations orientation of rational expectations modelling

to its logical extreme. It views the economy as determined by a set of optimizing agents

with rational expectations operating with access to fully-functioning markets.

While many now question the specific assumptions underlying the early RBC models,

the methodology has endured. Today’s DSGE model builders may view aspects of the

economy differently (e.g. the role of monetary policy) but they share the goal of motivating

all of their model’s relationships on the basis of optimizing behaviour by households and

firms. In these notes, we will set out a basic RBC model and discuss how the model’s

first-order conditions can be turned into a system of linear difference equations of the form

we know how to solve. We will also discuss some of the arguments for and against RBC

models.

The RBC Model Economy

The most basic RBC models assume perfectly functioning competitive markets. This means

that the outcomes generated by decentralized decisions by firms and households can be

replicated as the solution to a social planner problem. In the RBC model that we will

examine, the social planner wants to maximize

Et

[

∞∑

i=0

βi

(

C1−ηt+i

1 − η− aNt+i

)]

(1)

where Ct is consumption, Nt is hours worked, and β is the representative household’s rate

of time preference. The economy faces constraints described by

Yt = Ct + It = AtKαt−1N

1−αt (2)

Kt = It + (1 − δ)Kt−1 (3)

1“Time to Build and Aggregate Fluctuations,” Econometrica, November 1982, Volume 50, pages 1345-

1370. This paper was cited in the 2004 Nobel prize award given to Kydland and Prescott.

Phd Macro, 2007 (Karl Whelan) 2

and a process for the technology term At. To simplify some aspects of the derivations below,

I will assume that At does not trend over time, so our model economy has an average growth

rate of zero (changing this so the economy grows on average changes very little of what

follows). Specifically, we will assume that the technology term follows an AR(1) process of

the form

log At = (1 − ρ) log A∗ + ρ log At−1 + ǫt (4)

Objections to the RBC Approach

Before discussing how to solve these models, how to simulate them, and their properties, it

is worth acknowledging that a lot of people have a negative reaction to them. Some of the

objections regularly aired are as follows:

• Perfect Markets and Rational Expectations: The idea that the economy can character-

ized as a perfectly competitive market equilibrium solution describing the behaviour

of a set of completely optimizing rational individuals does not appeal to all. Surely

we know that markets are not always competitive and don’t always work well, and

surely people are not always completely rational in their economic decisions?

• Monetary and Fiscal Policy : RBC models exhibit complete monetary neutrality, so

there is no role at all for monetary policy, something which many people think is

crucial to understanding the macroeconomy. We haven’t put government spending

in this model, but if we did, the model would exhibit Ricardian equivalence, so the

effects of fiscal policy would be different from what most people imagine them to be.

• Skepticism about Technology Shocks: RBC models give primacy to technology shocks

(the ǫt shocks) as the source of economic fluctuations. But what are these shocks? Is it

really credible that all economic fluctuations, including recessions, are just an optimal

response to technology shocks? What are these technology shocks anyway—if we were

experiencing these big shocks wouldn’t we be reading about them in the newspapers?

Wouldn’t recessions have to be periods in which their is outright technological regress,

so that (for some reason) firms are using less efficient technologies than previously—is

this credible?

Phd Macro, 2007 (Karl Whelan) 3

Discussion of these Objections

There is some substance to most (though not all) of these criticisms. However, they should

not be seen as arguments against the usefulness of the RBC approach:

• The RBC model should be seen as a benchmark against which more complicated

models can be assessed. If a model with optimizing agents and instantaneous market

clearing can explain the business cycle, then can market imperfections such as sticky

prices really be seen as crucial to understanding macroeconomic fluctuations? To

explain why these imperfections might be important, one needs to understand the

limitations of the RBC model.

• The framework provided by RBC models can also be extended to account for a wide

range of factors not considered in the basic model discussed here. For instance, if we

allow for imperfect competition, then the decentralized market outcome cannot be

characterized as the outcome of a social planner problem. So, one needs to derive the

FOCs from separate modelling of the decisions of firms and households. But this is

easily done. In addition, other frictions such as sticky prices—which would allow for

monetary policy to have a role—can also be added.

• Technology shocks are probably a bit more important than one might think at first.

Growth theory teaches us that increases in TFP are the ultimate source of long-run

growth in output per hour. Is there any particular reason to think that these increases

have to take place in a steady trend-like manner? Put this way, random fluctuations

in TFP growth doesn’t seem so strange. Also, at least in theory, RBC models could

generate recessions without outright declines in technology. This can happen if the

elasticity of output with respect to technology is greater than one; then potentially,

a slowdown in TFP growth below trend could trigger a recession.

A Technical Comment

Before getting on with solving the model, I’ll make a brief technical comment about maxi-

mization techniques for problems like this; don’t worry if you don’t understand this. Tech-

nically, the best way to solve optimization problems like this that feature random variables

is called stochastic dynamic programming. I don’t have the time to teach these methods in

this class, so instead I will use Lagrangian methods, which I know you have seen. In doing

Phd Macro, 2007 (Karl Whelan) 4

this, I will assume that one can differentiate things of the form EtF (x) to give EtF′ (x).

One might wonder if this is ok, but it is. To see this consider the problem of choosing a

value of x to maximize the expected value of F (a, x) where a is random and can take on

the value ak with probability pk. This expected value is given by

G (x) = Et

N∑

k=1

pkF (ak, x) (5)

This is maximized by setting

G′ (x) = Et

N∑

k=1

pkF′ (ak, x) = EtF

′ (x) = 0 (6)

So, the FOC for maximizing EtF (x) is just EtF′ (x) = 0.

The Social Planner’s Problem

The Lagrangian for this problem can be written as

L = Et

[

∞∑

i=0

βi

{

C1−ηt+i

1 − η− aNt+i + λt

(

At+iKαt+i−1N

1−αt+i + (1 − δ)Kt+i−1 − Ct+i − Kt+i

)

}]

(7)

(For convenience, I have substituted the capital accumulation constraint into the resource

constraint.) Now this equation might look a bit intimidating. It involves an infinite sum,

so technically there is an infinite number of first-order conditions for current and expected

future values of Ct, Kt and Nt. To see that the problem is less hard than this makes it

sound, note that the time-t variables appear in this sum as

C1−ηt

1 − η− aNt + λt

(

AtKαt−1N

1−αt − Ct − Kt

)

+ βEt

(

λt+1At+1Kαt N1−α

t+1 + (1 − δ)Kt

)

(8)

After that, the time-t variables don’t ever appear again. So, the FOCs for these variables

consist of differentiating equation (8) with respect to the relevant variables and setting the

derivatives equal to zero. After that, the time t + n variables appear exactly as the time

t variables do, except that they are in expectation form and they are multiplied by the

discount rate βn. But the FOCs for the time t + n variables will be identical to those for

the time t variables. So differentiating equation (8) is enough to give us the equations that

describe the optimal dynamics at all times for this economy. These FOCs are

∂L

∂Ct

: C−ηt − λt = 0 (9)

Phd Macro, 2007 (Karl Whelan) 5

∂L

∂Kt

: βEt

(

αYt+1

Kt

λt+1 + (1 − δ)λt+1

)

− λt = 0 (10)

∂L

∂Nt

: −a + (1 − α) λtYt

Nt

= 0 (11)

∂L

∂λt

: AtKαt−1N

1−αt + (1 − δ)Kt−1 − Ct − Kt = 0 (12)

One trick that helps make this system a bit easier to understand is to define the marginal

value of an additional unit of capital next year as

Rt+1 = αYt+1

Kt

+ 1 − δ (13)

This will be the gross interest rate for this economy (i.e. the interest rate will be Rt+1 −1).

With this definition in hand, equation (10) can be written as

λt = βEt (λt+1Rt+1) (14)

and this can be combined with the equation (9) to give

C−ηt = βEt

(

C−ηt+1Rt+1

)

(15)

or, alternatively

βEt

((

Ct

Ct+1

)η

Rt+1

)

= 1 (16)

The Full Set of Model Equations

The RBC model can then be defined by the following six equations (three identities de-

scribing resource constraints and three FOCs)

Yt = Ct + It (17)

Yt = AtKαt−1N

1−αt (18)

Kt = It + (1 − δ)Kt−1 (19)

1 = βEt

((

Ct

Ct+1

)η

Rt+1

)

(20)

Yt

Nt

=a

1 − αC

ηt (21)

Rt = αYt

Kt−1+ 1 − δ (22)

as well as the process for the technology variable.

Phd Macro, 2007 (Karl Whelan) 6

Those of you who have been following me thus far might notice something funny about

the system given by equations (17) to (22). They are not a set of linear difference equa-

tions, but a mix of both linear and nonlinear equations: Haven’t I been saying that DSGE

modelling is all about sets of linear difference equations? In general, these nonlinear sys-

tems cannot be solved analytically. However, it turns out their solution can be very well

approximated by turning them into a set of linear equations in the deviations of the logs

of these variables from their steady-state values. This approach gives us a system that

expresses variables in terms of their percentage deviations from the steady-state paths in

which all real variables are growing at the same rate. The steady-state path is relevant

because the stochastic economy will, on average, tend to fluctuate around the values given

by this path. This method can be thought of as giving a system of variables that represents

the business-cycle component of the model.

Log-Linearization: What’s The Deal?

The deal is as follows. The particular economy that we are looking has a zero-growth

steady-state path, so for each real variable Xt there is a corresponding X∗ that is constant

on the steady-state path. We will use lower-case letters to define log-deviations of variables

from their steady-state values.

xt = log Xt − log X∗ (23)

The key to the log-linearization method is that every variable can be written as

Xt = X∗Xt

X∗= X∗ext (24)

Then, the big trick is that a first-order Taylor approximation of ext is given by

ext≃ 1 + xt (25)

So, we can write variables as

Xt ≃ X∗ (1 + xt)

The second trick is that when one has variables multiplying each other such as

XtYt ≃ X∗Y ∗ (1 + xt) (1 + yt) (26)

you set terms like xtyt = 0 because we are looking at small deviations from steady-state and

multiplying these small deviations together one gets a term close to zero. So, the log-linear

approximation for these terms is

XtYt ≃ X∗Y ∗ (1 + xt + yt) (27)

Phd Macro, 2007 (Karl Whelan) 7

That’s it. Substitute these approximations for the variables in the model, lots of terms end

up canceling out, and when you’re done you’ve got a system in the deviations of logged

variables from their steady-state values. The paper on the reading list by Harald Uhlig

discusses this stuff in a bit more detail and provides more examples.

Log-Linearization: Example 1

I will not go through the details of the log-linearization of all six of our model equations

from (17) to (22). Instead, I will describe the log-linearization of (17) and (18) and then

leave the other four equations for you to go through in your problem set. So, start with

equation (17). This equation can also written as

Y ∗eyt = C∗ect + I∗eit (28)

Using the first-order approximation, this becomes

Y ∗ (1 + yt) = C∗ (1 + ct) + I∗ (1 + it) (29)

Note, though, that the steady-state terms must obey identities so

Y ∗ = C∗ + I∗ (30)

Canceling these terms on both sides, we get

Y ∗yt = C∗ct + I∗it (31)

which we will write as

yt =C∗

Y ∗ct +

I∗

Y ∗it (32)

This is the log-linearized version of the equation that we will work with.

Log-Linearization: Example 2

Now consider equation (18). This can be re-written in terms of steady-states and log-

deviations as

Y ∗eyt = (A∗eat) (K∗)α eαkt (N∗)1−α e(1−α)nt (33)

Again, use the fact the steady-state values obey identities so that

Y ∗ = A∗ (K∗)α (N∗)1−α (34)

Phd Macro, 2007 (Karl Whelan) 8

So, canceling gives

eyt = eateαkte(1−α)nt (35)

Using first-order Taylor approximations, this becomes

(1 + yt) = (1 + at) (1 + αkt) (1 + (1 − α) nt) (36)

Ignoring cross-products of the log-deviations, this simplifies to

yt = at + αkt + (1 − α) nt (37)

The Full Log-Linearized System

Once all the equations have been log-linearized, we have a system of seven equations of the

form

yt =C∗

Y ∗ct +

I∗

Y ∗it (38)

yt = at + αkt + (1 − α) nt (39)

kt =I∗

K∗it + (1 − δ) kt−1 (40)

nt = yt − ηct (41)

ct = Etct+1 −1

ηEtrt+1 (42)

rt =

(

α

R∗

Y ∗

K∗

)

(yt − kt−1) (43)

at = ρat−1 + ǫt (44)

We are nearly ready to put the model on the computer. However, notice that three of the

equations have coefficients that are values relating to the steady-state path. These need to

be calculated.

Steady-State Coefficients

This turns out to be pretty easy. Start with the steady-state interest rate. We have

1 = βEt

((

Ct

Ct+1

)η

Rt+1

)

(45)

Because we have no trend growth in technology in our model, the steady-state features

consumption, investment, and output all taking on constant values. Thus, in steady-state,

we have C∗

t = C∗

t+1 = C∗, so

R∗ = β−1 (46)

Phd Macro, 2007 (Karl Whelan) 9

Next, we have

Rt = αYt

Kt−1+ 1 − δ (47)

So, in steady-state, we haveY ∗

K∗=

β−1 + δ − 1

α(48)

Together with the steady-state interest equation, this tells us that

α

R∗

Y ∗

K∗= αβ

(

β−1 + δ − 1

α

)

= 1 − β (1 − δ) (49)

which is the steady-state value required for equation (43). Next, we use the identity

Kt = It + (1 − δ)Kt−1 (50)

and the fact that in steady-state we have K∗

t = K∗

t−1 = K∗, to give

I∗

K∗= δ (51)

which is required in equation (40). This can then be combined with the previous steady-

state ratio to giveI∗

Y ∗=

I∗

K∗

Y ∗

K∗

=αδ

β−1 + δ − 1(52)

and obviouslyC∗

Y ∗= 1 −

αδ

β−1 + δ − 1(53)

which gives us the steady-state ratios in equation (38).

The Final System

Using these steady-state identities, our system becomes

yt =

(

1 −

αδ

β−1 + δ − 1

)

ct +

(

αδ

β−1 + δ − 1

)

it (54)

yt = at + αkt + (1 − α) nt (55)

kt = δit + (1 − δ) kt−1 (56)

nt = yt − ηct (57)

ct = Etct+1 −1

ηEtrt+1 (58)

rt = (1 − β (1 − δ)) (yt − kt−1) (59)

at = ρat−1 + ǫt (60)

Phd Macro, 2007 (Karl Whelan) 10

This is written in the standard format that allows us to apply solution algorithms such as

the Binder-Pesaran program to obtain a reduced-form solution, and thus simulate the model

on the computer.2 One thing to recall is that all of these coefficients can be interpreted as

elasticities. Also, impulse response functions can be interpreted as describing the percentage

responses to one percent shocks.

RBC Models Generate Cycles

Figure 1 simulates our model. It uses parameter values intended for analysis of quarterly

time series: α = 13 , β = 0.99, δ = 0.015, ρ = 0.95, and η = 1.3 RATS programs to

generate charts of this type are available on the website, so if you wanted, you could work

with the model yourself. The figure demonstrates the main successful feature of the RBC

model: It generates actual business cycles and they don’t look too unrealistic. In particular,

reasonable parameterizations of the model can roughly match the magnitude of observed

fluctuations in output, and the model can match the fact that investment is far more volatile

than consumption.

Deficiencies of RBC Models

Despite this success, RBC models have still come in for some criticism. One reason is

that they have not quite lived up to the hype of their early advocates. Part of that hype

stemmed from the idea that RBC models contained important propagation mechanisms

for turning technology shocks into business cycles: Increases in technology induced extra

output through higher capital accumulation and by inducing people to work more. In other

words, some of the early research suggested that even in a world of iid technology levels, one

would expect RBC models to still generate business cycles. However, Figure 2 shows that

output fluctuations in this model follow technology fluctuations quite closely: This shows

that these additional propagation mechanisms are quite weak.

Cogley and Nason (1995) note another fact about business cycles that the RBC model

2Most of the textbook treatments of RBC models discuss solving it with the “method of undetermined

coefficients” as discussed in the paper on the reading list by John Campbell. This involves guessing the

form of the solution and then plugging that solution back into the equation to determine the coefficients.

Campbell’s paper is great—it has lots of useful insights into the RBC model—but this generally requires

far too much algebra to be practical.

3This last value might look a bit funny because technicallyC

1−η

t+i

1−ηis not defined at this value. However,

this function tends towards log Ct as η tends to one, and that is what is meant by this.

Phd Macro, 2007 (Karl Whelan) 11

does not match. Output growth is positively autocorrelated (not very—autocorrelation

coefficient of 0.34—but significantly so in a statistical sense). But RBC models do not

generate this pattern: See Figure 3. They can only do so if one simulates a technology

process that has a positively autocorrelated growth rate.

Cogley and Nason relate this back to the IRFs generated by RBC models. Figure 4

shows the responses of output, consumption, investment, and hours to a unit shock to

ǫt. Figure 5 highlights that the response of output to the technology shock pretty much

matches the response of technology itself. Cogley-Nason argue that one needs instead to

have “humped-shaped” responses to shocks—a growth rate increase needs to be followed

by another growth rate increase—if a model is to match the facts about autocorrelated

output growth. The responses to technology shocks do not deliver this. In addition, while

we don’t have other shocks in the model (e.g. government spending shocks), Cogley-Nason

point out that RBC models don’t generate hump-shaped responses for these either.

A more recent critique relates to the response of hours to the technology shock. The

VAR paper by Gali that we discussed a few weeks ago argues that the data point to

technology shocks actually having a negative effect on hours worked, not a positive one.

This argues against the model’s interpretation of labor market fluctuations. Others have

also argued that the model’s assumption that hours worked are highly responsive to real

wages does not seem to match the microeconomic evidence on labour supply.

There are currently a number of branches of research aimed at fixing the deficiencies of

the basic RBC approach. Some of them involve putting extra bells and whistles on the basic

market-clearing RBC approach: Examples include variable utilization, lags in investment

projects, habit persistence in consumer utility. The second approach is to depart more

systematically from the basic RBC approach by adding rigidities such as sticky prices and

wages. Some papers do both. I plan to spend some of the next few weeks looking at some

of these models.

Output Investment Consumption

Figure 1RBCs Generates Cycles with Volatile Investment (Simulation)

25 50 75 100 125 150 175 200-15

-10

-5

0

5

10

15

Output Technology

Figure 2But the Cycles Rely Heavily on Technology Fluctuations (Simulation)

25 50 75 100 125 150 175 200-3.6

-2.7

-1.8

-0.9

-0.0

0.9

1.8

2.7

Output Growth

Figure 3RBCs Do Not Generate Autocorrelated Growth (Simulation)

25 50 75 100 125 150 175 200-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Output Consumption Investment Hours

Figure 4RBC Model: Impulse Responses to Technology Shock

5 10 15 20 25 30 35 40 45 50-1.6

0.0

1.6

3.2

4.8

6.4

8.0

9.6

Output Technology

Figure 5RBC Model: Impulse Responses to Technology Shock

5 10 15 20 25 30 35 40 45 500.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25