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
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