Lee Ohanian
Economics 402
UCLA
1 Notes on Business Cycles
Business cycles are persistent movements in macroeconomic variables above or
below their trend values. The trend is usually defined as either a log-linear
trend, in which the cyclical component is the residual of a regression of the log
of variable Y on time and a constant term, where the lower case letter denotes
the log of the variable, and yct is the cyclical component of the yt :
yct = yt − α− βt
To make this operational, all we need to do is to fit a regression of yt on a
constant term, and the time trend, in which t = 1, 2, 3, …Thus, the term yct is
just the residual from this regression:
yt = α+ βt+ εt, yct ≡ εt (1)
There is an alternative procedure for detrending that is often used in busi-
ness cycle analysis, which is known as “Hodrick-Prescott”detrending. You do
not need to memorize this procedure for examination purposes. This is quite
similar to linear detrending, but it allows for some small changes in the trend.
Specifically, the trend will not a line, as it is with linear detrending, but rather
it will be a curve. We will see this a bit later. The Hodrick-Prescott trend is
given by:
yHPct = yt − τ t,
where τ t is the minimizer of:
T∑
t=1
(yt − τ t)2 + λ
T−1∑
t=2
[(τ t+1 − τ t)− (τ t − τ t−1)]2
and where λ is a parameter, whose value is given. Hodrick and Prescott suggest
a value for λ of 1,600 for quarterly data. For annual data, the value is lower.
Note that as λ tends to infinity, the trend component tends to a linear trend, as
earlier in the notes. Frequent values for annual data range from around 6 to 100.
There are several programs written in Matlab that performs this minimization.
This information is presented for completeness. You wont be examined on this
material about the Hodrick-Prescott Filter. The simple detrending approach
using a time trend and least squares will give results that are not too different.
1
张熠华�
张熠华�
张熠华�
张熠华�
The paper by Kydland and Prescott, ”Business Cycles: Real Facts and a
Monetary Myth”, which is posted on the website, shows the statistical proper-
ties of macroeconomic time series that have been detrended using the Hodrick-
Prescott approach. Many of these detrended time series are well approximated
as low-order difference equations with the largest root that is close to 1. For ex-
ample, the following first order autoregressive process reasonably approximates
many detrended economic time series, in which the data mean has been removed:
yct = ρyct−1 + εt, E(εt) = 0, E(ε
2
t ) = σ
2
with ρ typically having a value between 0.8 – 1.0. This is called autoregressive
because the value of the random variable today depends on its value yesterday,
hence the term “auto”, which means self, and regressive, which means looking
backwards in time. It is called first order because there is only 1 lag. The nth
order autoregressive (AR) process is given by
yt =
n∑
n=i
ρiyt−i + εt (2)
The autoregressive process is an example of a stochastic process. Another sto-
chastic process that is used commonly in business cycle studies is the moving
average process. The nth order moving average (MA) process is given by:
yt =
n∑
n=i
θiεt−i + εt (3)
This is what you need to know about stochastic processes for this course. If you
are interested, I can give you more information.
The first order autoregressive process can be written as an infinite order an
infinite order moving average process. To see this, note that we begin with:
yt = ρyt−1 + εt (4)
and moving back one period in time, we have
yt−1 = ρyt−2 + εt−1 (5)
Next, substitute out for yt−1 in the first equation above, using the fact that
yt−1 = ρyt−2 + εt−1. After substituting, we get:
yt = εt + ρεt−1 + ρ
2yt−2 (6)
Next, substitute out for yt−2 the exact same way, and just keep moving
backwards in time. This process is called “backward substitution”. If we keep
doing this, we get:
yt = εt + ρεt−1 + ρ
2εt−1 + ρ
3εt−3 + ρ
4εt−4…
2
张熠华�
张熠华�
张熠华�
张熠华�
张熠华�
张熠华�
张熠华�
After doing this substitution, note that the variable yt is the accumulated
weighted sum of all of the shocks. If the coeffi cient ρ is less than one, then
the affect of the shocks die out over time. If ρ is equal to 1, then the variable
yt is just the sum of the accumulated shocks, which do not die out.
Figure 1 in their paper shows an example of how this type of process with
ρ = 0.95 can generate movements in a random variable that look like a business
cycle, with persistent deviations above and below the horizontal axis. Thus, one
key property of business cycle fluctuations is that they are persistent: when the
economy is above trend, it will tend to remain above trend for some time, and
similarly when the economy is below trend, it will be expected to remain below
for some time.
Another key property of business cycles is that many macroeconomic time
series, particularly those variables that are quantity variables, are positively cor-
related. In particular, the major output components (consumption, investment,
GDP), production inputs, and productivity are all positively related. The final
aspect of the statistical behavior of business cycles we will consider is volatil-
ity. Noteworthy features include: (1) investment is substantially more volatile
than output, (ii) consumption is substantially less volatile than output, and (iii)
hours are nearly as volatile as output, (iv) productivity is quite volatile. We
will see this in the Kydland-Prescott paper.
2 Business Cycles in an Optimizing Model
This section develops a model of business cycles, using the optimal growth model
as a foundation. As a ”warm-up”exercise, let’s see if we can use the business
cycle statistics we have just examined to determine whether certain classes of
business cycle models are likely to be successful.
First, let’s consider ”demand shocks”, which are probably the most popular
class of business cycle shocks that are cited in the popular press and among
government agencies, such as central banks, finance ministeries, and treasury
departments.
The Keynesian model is a demand-shock based model. We will see that
in the standard growth model, this class of models by itself cannot account
for these facts. By demand shocks, we mean exogenous shifts in the demand
for goods and services. These demand shocks could be exogenous changes in
government spending, or exogenous changes in preferences. The reason that
demand shocks cannot account for these data within the standard neoclassical
model is because they predict countercyclical labor productivity, rather than
procyclical labor productivity. Specifically, recall labor productivity in the
Cobb-Douglas production function, without any technological shocks or changes:
Yt
Ht
= A(1− θ)
(
Kt
Ht
)θ
Since capital is fixed, and there is no technological change, then the increase
in hours worked due to a demand shock that raises output necessarily lowers
3
张熠华�
张熠华�
张熠华�
labor productivity. Empirically, however, we observe the opposite. Labor pro-
ductivity is higher during booms, and low during recessions. Thus, demand
shocks, by themselves, cannot match this fact about fluctuations.
We will now develop a model of business cycles that can match the pro-
ductivity data. We will start using the optimal growth model, with the Cobb-
Douglas production function, and add in one modeling feature, which will be a
productivity shock:
Yt = ztK
θ
t (AtHt)
1−θ (7)
The productivity shock is denoted as zt. The specification of this shock is an
autoregressive process that is log-normal:
ln(zt) = ρ ln(zt−1) + εt, εt ∼ N(0, σ2), 0 < ρ < 1 (8) This introduces random productivity in the model. Note that it is common to model the productivity shock in logarithmic form to insure that the shock never becomes negative. First, let’s see how we can measure z. All we need to do is to construct z by first constructing per-capita real GDP, per capita real capital stock, and per-capita labor input. Then we take the linear trend out of log per capita real GDP and log per capita real capital. We then form the following expression for ln(zt) in which the GDP and capital are logged and detrended, and labor is logged, and we use the income shares of 1/3 and 2/3: ln(zt) = yt − 1/3 ∗ kt − 2/3 ∗ lt (9) First, let’s take a look at a two period model with preferences that allow us to construct a solution to the model. Assume that kt is given. p Ut = max{ct − h2t 2 + β(ct+1 − h2t+1 2 )} (10) The resource constraints are given by: ztk θ t h 1−θ t + (1− δ)kt = ct + kt+1 (11) zt+1k θ t+1h 1−θ t+1 + (1− δ)kt+1 = ct+1 (12) Form the Lagrangian and solve for the first order conditions for ct, ht, ct+1, ht+1 and kt+1. To conserve space, note that I am using Ut from above: L = max{Ut+λ1[ztkθt h 1−θ t +(1−δ)kt−ct+kt+1]+λ2[zt+1k θ t+1h 1−θ t+1+(1−δ)kt+1−ct+1]} (13) 4 The first order conditions in terms of differentiating with respect to the choice variables ct, ct+1, ht, ht+1, and kt+1 are: 1 = λ1 (14) β = λ2 (15) φht = (1− θ)zt ( kt ht )θ (16) φht+1 = (1− θ)zt+1 ( kt+1 ht+1 )θ (17) 1 = β[zt+1θ ( ht+1 kt+1 )1−θ + 1− δ] (18) Let’s see how changes in productivity affect decisions. Suppose that the social planner knows both zt and zt+1, and assume that the mean of z = 1. Case 1 - zt is different from one, zt+1 = 1. Let’s look at the first order condition for choosing labor. Doing a bit of algebra and re-arranging terms, we get: ht = [(1− θ)ztkt] 1 1+θ (19) We can see clearly how z affects the work decision. If zt is high, then people work more - ∂ht ∂zt > 0. This is because productivity is high, which incentivizes
more work.
Note that because we chose preferences that are linear in consumption in
both periods, then the decision to save is not affected by zt. We will see later
that variation in zt will motivate changes in savings with preferences that are
not linear. However, we will see a bit later that variation in zt+1 will indeed
motivate changes in savings decisions in this environment.
Case 2 – zt+1 is different from one, zt = 1.
Note that we also have that ∂ht+1
∂zt+1
> 0, since ht+1 = [(1− θ)zt+1kt+1]
1
1+θ .
Thus, higher productivity in period t+1 also incentivizes more work.
Next, let’s see how zt+1 affects savings decisions. To do this, we look at the
first order condition for kt+1 :
1 = β[zt+1θ
(
ht+1
kt+1
)1−θ
+ 1− δ] (20)
We already know that high zt+1 leads to high ht+1. So let’s substitute out
for ht+1 in the first order condition for savings. Recall
5
ht+1 = [(1− θ)zt+1kt+1]
1
1+θ (21)
Now, substitute that equation in for ht+1 in the savings first order condition:
1 = β[zt+1θ
(
[(1− θ)zt+1kt+1]
1
1+θ
kt+1
)1−θ
+ 1− δ] (22)
Simplifying, we get:
1 = βθ(1− θ)
1−θ
1+θ z
1
1+θ
t+1 k
−θ(1−θ)
1+θ
t+1 (23)
Next, isolating the variables kt+1, we get:
kt+1 = [βθ(1− θ)
1−θ
1+θ z
1
1+θ
t+1 ]
1+θ
θ(1−θ) (24)
Note that a high zt+1 relative to its normal value means that kt+1 is higher
than its normal value. So high productivity in the future motivates the planner
to save more and accumulate more capital. This allows society to take advantage
of future high productivity by having more productive inputs in place.
Next, let’s see how changes in preferences affect allocations. Modify the
utility function to be:
Ut = max{ct −
h2t
2
+ β(αt+1ct+1 −
h2t+1
2
)} (25)
Now, households care about consumption differently between the two peri-
ods. Suppose that αt+1 > 1. This means that marginal utility in period t+1
will be relatively high compared to marginal utility in period t.
Let’s see how this affects the economy. The first order conditions become:
1 = λ1 (26)
βαt+1 = λ2 (27)
φht = (1− θ)zt
(
kt
ht
)θ
(28)
φht+1 = (1− θ)zt+1αt+1
(
kt+1
ht+1
)θ
(29)
1 = βαt+1[zt+1θ
(
ht+1
kt+1
)1−θ
+ 1− δ] (30)
The equations that change are the conditions for choosing how much to work
in period t+1, and how much to save for period t+1. In terms of work effort
in period t+1, note that the higher marginal utility of consumption motivates
more work, all other things equal. We can see this as a high value of αt+1
6
raises the right hand side of the equation. This is easy to understand as follows.
People like to consume a lot at the margin in period t+1, so one way to achieve
higher consumption in period t+1 is to work more.
Higher marginal utility in period t+1 also motivates people to save more. In
particular, look at the right hand side of the first order condition for saving. A
high value of αt+1 raises the right hand side of the equation by itself. Moreover,
we already know that ht+1 will be bigger. This means that kt+1 will be bigger
in order to satisfy the equality. People like to consume a lot in period t+1.
People will also work more in period t+1, so both the higher marginal utility
of consumption, plus more labor motivates extra capital accumulation. This
larger amount of capital also supports the higher level of consumption that the
consumer wishes to have.
As an exercise, try to analyze what happens if αt+1 is less than one. You
should see that this will motivate less work in period t+1 and less capital accu-
mulation than would occur if αt+1 = 1.
7