Chapter 28
Business cycle fluctuations
This chapter presents stylized facts and basic concepts relating to business
cycle fluctuations. The next chapters go more into depth with specific busi-
ness cycle theories.
The term business cycles refers to the empirical phenomenon of economy-
wide fluctuations in output and employment about trend, observed in in-
dustrialized market economies. These ups (expansions) and downs (contrac-
tions) in aggregate economic activity are hump-shaped rather than saw-tooth
shaped. The word “cycle” should not be taken literally, since the sequence
of expansions and contractions is not periodic like sinus waves. But the se-
quence shows many statistical regularities and these are what business cycle
analysts focus on.
28.1 Some business cycle facts
Compared with “white noise fluctuations”, business cycle fluctuations are
characterized by composite stochastic regularities. In a short list we empha-
size the following regularities displayed by time series data:
1. GDP and employment exhibit fluctuations about trend (whether the
trend is best described as stochastic or deterministic is a recurrent
theme in econometric time series analysis)
2. The ups and downs (expansions and contractions) exhibit quasi-persistence
(duration) in that positive deviations from trend tend to be maintained
over several periods and negative deviations from trend similarly (pos-
itive auto-correlation).
3. The ups and downs tend to be hump-shaped rather than saw-tooth
shaped (amplification).
1013
1014 CHAPTER 28. BUSINESS CYCLE FLUCTUATIONS
4. The fluctuations are recurrent, but neither periodic nor easily pre-
dictable. The distance from peak to peak may be, say, 4-12 years.
5. The fluctuations exhibit systematic co-movement across production
sectors, GDP components, and countries. Some facts that have played
a central role for the theoretical debate are:
(a) Employment (aggregate labor hours) is procyclical, i.e., varies in
the same direction as GDP, and fluctuates almost as much as GDP.
(b) Aggregate consumption and employment are markedly positively
correlated.
(c) Real wages are weakly procyclical and do not fluctuate much.
Some of the regularities identified may only be valid for a subset of coun-
tries, depending on the structural characteristics of these. For example Fig.
28.1 shows that unemployment in Europe as well as the US fluctuates con-
siderably. Only in the US, however, has unemployment appeared stationary
since the early 1970s.
The next section gives a list of definitions of terms often used by business
cycle theorists.
28.2 Key terms from the business cycle vo-
cabulary
Impulse versus response. The “impulse” is a disturbance to the economic
system coming “from the outside”. Is synonymous with a “shock” to an
exogenous variable (an unanticipated shift in its value). The “response”
refers to the reaction of the economic system, i.e., the effect on endogenous
variables.
Propagation and propagation mechanism. “Propagation” refers to the
spreading of effects of the impulse through the economic system (synonymous
with “dissemination”, “transmission” or “proliferation”). Then, “propaga-
tion mechanism” is just the economic mechanism involved in this spreading.
The propagation mechanism can lead to amplification, persistence and
co-movement :
Amplification is present when an per cent deviation (from normal) of an
exogenous variable results in a more than per cent deviation (from normal)
of an endogenous variable. Is more or less synonymous with “magnification”,
“multiplier effect” or “blow up effect”.
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28.2. Key terms from the business cycle vocabulary 1015
0
2
4
6
8
10
12
1970 1975 1980 1985 1990 1995 2000 2005
Western Europe Denmark USA Eurozone
Figure 28.1: The rate of unemployment in Denmark, Western Europe, the Eu-
rozone and the United States, 1970-2005. Note: Unemployment is measured as
the number of unemployed relative to labour force. Western Europe comprises the
EU-15 as well as Norway, Switzerland and Iceland. Germany is only included after
the reunification 1991. Source: OECD, Economic Outlook.
Persistence refers to effects on endogenous variables along another di-
mension, namely the time dimension. A shock has “persistent” effects to the
extent that the effects last long. Is synonymous with durability of the effect.
Is often measured by the auto correlation coefficient calculated from the time
series of the endogenous variable. Sometimes the shock itself is said to be
persistent, usually meaning that there is a relatively permanent change in
an exogenous variable. It is thus important to be aware that the distinction
between “temporary” and “persistent” may refer to either the effect of the
shock or the shock itself. Table 1 gives a reminder, where also the intermedi-
ate possibility, gradually “fading”, is included. The border line between the
intermediate category end the end categories is not sharp.
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1016 CHAPTER 28. BUSINESS CYCLE FLUCTUATIONS
Table 1 Glossary concerning shocks and their effects
Shock type
Effect on dependent variable Temporary Persistent Permanent
Temporary
Persistent
Permanent
Co-movement refers to the presence of significant correlation between two
or more de-trended variables (usually in logs).
Finally, volatility usually refers to the standard deviation (sometimes
variance) of the deviations of a variable from its trend value. Fixed capital
investment is much more volatile than GDP whereas consumption is consid-
erable less volatile.
28.3 A quick glance at the Great Recession
and its aftermath
Some data on labor market flows in the U.S. published by the Bureau of
Labor Statistics is shown in the figures 28.2 - 28.4. The terminology used
is the following: total separations equal the sum of quits and layoffs and
discharges, quits being separations on the initiative of the worker and layoffs
and discharges being separations initiated by the firm. Large fluctuations in
employment are envisaged. The shaded areas in the figures indicate periods
of recession as diagnosed by the NBER (National Bureau of Economic Re-
search). The NBER defines an economic recession as: “a significant decline
in economic activity spread across the economy, lasting more than a few
months, normally visible in real GDP, real income, employment, industrial
production, and wholesale-retail sales”.1 It is noteworthy that after the 2008-
2009 recession (the “Great Recession”) the trough level is lower than it was
after the dot.com-bubble 2001 recession .
At least two different stories could in principle explain this sharp fall
in employment.2 One is a “Schumpeterian story” about reallocation of la-
bor from old to new industries due to technological change. The other is a
“Keynesian story” about an overall fall in aggregate demand triggered by a
1A simpler definition, popular in the press, is that a recession is present if in two
consecutive quarters real GDP falls.2Krugman, New York Times, Dec. 11, 2010.
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
28.3. A quick glance at the Great Recession and its aftermath 1017
financial crisis. A believer of the Schumpeterian story would expect total sep-
arations, quits, and hiring to rise during the recession, as workers move from
obsolete industries to blossoming industries. The figures 28.2 and 28.3 in-
dicate the opposite: total separations, quits, and hiring behave procyclically
not countercyclically.
A believer of the Keynesian story would expect layoffs and discharges
to rise and hiring to fall during the recession, as firms generally need fewer
workers to satisfy the slack demand. In addition, this story predicts that quits
should fall, as there is a perception that vacant jobs are scarce. These three
predictions are confirmed by the figures. The combination of a rise in layoffs
and discharges and a fall in quits implies that the direction in which total
separations move is ambiguous according to the Keynesian story. Fig. 28.2
indicates that total separations fell during both the dot.com-bubble recession
in 2001 and the Great recession 2008-2009; so we can conclude that the fall
in quits dominated. Moreover, for the whole decade Fig. 28.3 suggests a
negative correlation between quits and layoffs and discharges.
In Fig. 28.4 we see a Beveridge curve for the U.S. based on observations
over a decade. The variable drawn along the horizontal axis in Fig. 28.4
is the unemployment rate in different months since year 2000 (number of
unemployed people as a percentage of the labor force). The variable drawn
along the vertical axis in the figure is the “job openings rate” in the same
months; an alternative name for this variable is the vacancy rate (number of
vacant jobs as a percentage of the labor force). As expected, the Beveridge
curve (so named after the British economist William Henry Beveridge, 1879-
1963) is negatively sloped. In a boom, unemployment is low and vacancies
plenty because recruitment is difficult, as few workers are searching for a
job. In a slump unemployment is high and the vacancy rate low because
recruitment is easy, as many workers are searching for a job. In this way,
the economy’s position on the downward sloping Beveridge curve can be
interpreted as reflecting the state of the business cycle. Indeed, Fig. 28.4
shows that from the start of the recent recession in December 2007 until
October 2009, the economy moved down the curve as the vacancy rate fell
and “layoffs and discharges” rose.
An outward shift of the Beveridge curve is a sign of reduced matching
efficiency in the labor market. Such a mismatch phenomenon can be due to
fast technological and structural change. Firms in the new industries have
vacant jobs but it is hard to find appropriate workers. Since October 2009,
the economy has moved slightly up and to the left. This is a sign of increased
mismatch. On the other hand, as Barlevy (2011) concludes and the figure
suggests, increased mismatch can account for only 2 of the 5 percentage point
increase in the unemployment rate since December 2007. So in his Nobel
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1018 CHAPTER 28. BUSINESS CYCLE FLUCTUATIONS
Figure 28.2
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28.3. A quick glance at the Great Recession and its aftermath 1019
Figure 28.3
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1020 CHAPTER 28. BUSINESS CYCLE FLUCTUATIONS
Figure 28.4
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
28.4. Conclusion 1021
laureate lecture, Dale Mortensen (2011) concluded: “The real problem is
that demand for goods and services has not recovered because real interest
rates have remained too high”.
28.4 Conclusion
In the next chapters we consider different theoretical approaches to the ex-
planation of business cycle regularities.
28.5 Literature notes
Articles in Handbook of Macroeconomics (1999) and for example the macro-
economics textbook by Abel and Bernanke (2001) describe in more detail the
empirical regularities that characterize business cycle fluctuations, including
both the direction and the timing of the cyclical behavior of economic vari-
ables.
28.6 Exercises
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1022 CHAPTER 28. BUSINESS CYCLE FLUCTUATIONS
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
Chapter 29
The real business cycle theory
Since the middle of the 1970s two quite different approaches to the explana-
tion of business cycle fluctuations have been pursued. We may broadly clas-
sify them as either of a New Classical or a Keynesian orientation. The New
Classical school attempts to explain output and employment fluctuations as
movements in productivity and labor supply. The Keynesian approach at-
tempts to explain them as movements in demand and the degree of capacity
utilization.
Within the New Classical school the monetary mis-perception theory of
Lucas (1972, 1975) was the dominating approach in the 1970s. We described
this approach in Chapter 26. The theory came under serious empirical attack
in the late 1970s.1 From the early 1980s an alternative approach within
New Classical thinking, the Real Business Cycle theory, gradually took over.
This theory (RBC theory for short) was initiated by Finn E. Kydland and
Edward C. Prescott (1982) and is the topic of this chapter.2 Other major
contributions include Long and Plosser (1983), Prescott (1986), and Plosser
(1989).
The shared conception of New Classical approaches to business cycle
analysis is the that economic fluctuations can be explained by adding sto-
chastic disturbances to the neoclassical framework with optimizing agents, ra-
tional expectations, and market clearing under perfect competition. Output
and employment are seen as supply determined, the only difference compared
with the standard neoclassical growth model being that there are fluctuations
around the growth trend. The fluctuations are not viewed as deviations from
a Walrasian equilibrium, but as a constituent part of a moving stochastic
Walrasian equilibrium. Whereas in Lucas’ monetary mis-perception theory
1For a survey, see Blanchard (1990).2In 2004 they were awarded the Nobel prize, primarily for their contributions in two
areas: policy implications of time inconsistency and quantitative business cycle research.
1023
1024 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
from the 1970s the driving force were shocks to the money supply, the RBC
theory is based on the idea that economic fluctuations are triggered primar-
ily by recurrent technology shocks and other supply shocks. In fact, money
is completely absent from the RBC models of the 1980s. The fluctuations
in employment reflect fluctuations in labor supply triggered by real wage
movements reflecting shocks to marginal productivity of labor. Government
intervention with the purpose of stabilization is seen as likely to be counter-
productive. Given the uncertainty due to shocks, the market forces establish
a Pareto optimal moving equilibrium. “Economic fluctuations are optimal
responses to uncertainty in the rate of technological change”, as Edward
Prescott claims (Prescott 1986).3
Below we describe the basics of the RBC model. It is this type of model
we have in mind in this text when speaking of “RBC theory”. It constitutes
a subclass of what later has become known as Dynamic Stochastic General
Equilibrium models, DSGE models. This name refers to a broader class of
quantitative models, including models with an emphasis on money, nominal
price stickiness, and demand shocks.
29.1 A simple RBC model
The RBC theory is a non-monetary Ramsey growth model in discrete time to
which is added exogenous recurrent productivity shocks. The presentation
here is close to King and Rebelo (1999), available in Handbook of Macro-
economics, vol. 1B, 1999. As a rule, our notation is the same as that of
King and Rebelo, but there will be a few exceptions in order not to diverge
too much from the general notational principles in this text. The notation
appears in Table 29.1. The most precarious differences in comparison with
King and Rebelo are that we use in our customary meaning as a utility
discount rate and for elasticity of marginal utility of consumption.
The firm
There are two categories of economic agents in the model: firms and house-
holds; the government sector is ignored. First we describe the firm.
3The “economic philosophy” behind the RBC theory in the sense the term is used here,
was proclaimed in Prescott (1986).
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29.1. A simple RBC model 1025
Table 29.1. Notation
Variable King & Rebelo Here
Aggregate consumption same
Deterministic technology level same
Growth corrected consumption ≡ same
Growth corrected investment ≡ same
Growth corrected output ≡ same
Growth corrected capital ≡ same
Aggregate employment (hours) same
Aggregate leisure (hours) ≡ 1− same
Effective capital intensity
≡
Real wage
Technology-corrected real wage ≡
Real interest rate from end
period to end period + 1 +1Auto-correlation coefficient in
technology process
Discount factor w.r.t. utility 11+
Rate of time preference w.r.t. utility 1− 1
Elasticity of marginal utility of cons.
Elasticity of marginal utility of leisure same
Elasticity of output w.r.t. labor same
Steady state value of ∗
The natural logarithm log same
Log deviation of from steady state value ≡ log
≡ log ∗
Log deviation of from steady state value ≡ log
≡ log
∗
Technology
The representative firm has the production function
= ( ) (29.1)
where and are input of capital and labor in period while is
the exogenous deterministic “technology level”, and represents an exoge-
nous random productivity factor. The production function has constant
returns to scale and is neoclassical (i.e., marginal productivity of each fac-
tor is positive, but decreasing in the same factor). In applications, often a
Cobb-Douglas function is used.
It is assumed that grows at a constant rate, − 1 i.e.,+1 = 1 (29.2)
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1026 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
where is the deterministic technology growth factor. The productivity
factor is a stochastic variable which is assumed to follow a process of the
form
= ∗1−(−1)
so that log is an AR(1) process:
log = (1− ) log∗ + log−1 + 0 ≤ 1 (29.3)
The last term, represents a productivity shock which is assumed to be
white noise with variance 2.4 The auto-correlation coefficient measures
the degree of persistence over time of the effect on log of a shock. If = 0
the effect is only temporary; if 0 there is some persistence. The uncon-
ditional expectation of log is equal to log∗ (which is thus the expected
value “in the long run”). The shocks, may represent accidental events
affecting productivity, perhaps technological changes that are not sustain-
able, including sometimes technological mistakes (think of the introduction
and later abandonment of asbestos in the construction industry). Negative
realizations of the noise term may represent “technological regress”. But
it need not, since moderate negative values of are consistent with overall
technological progress, though temporarily below the trend represented by
the deterministic growth of
The reason that we said “not sustainable” is that sustainability would
require = 1 which conflicts with (29.3). Yet = 1 which turns (29.3) into
a random walk with drift, would correspond better to our general conception
of technological change as a cumulative process. Technical knowledge is cu-
mulative in the sense that a technical invention continues to be known and
usable. But in the present version of the RBC model this cumulative part of
technological change is represented by the deterministic trend in (29.2)5 It
remains somewhat vague what the stochastic really embodies. A broad in-
terpretation includes abrupt structural changes, closures of industries, shifts
in legal and political systems, harvest failures, wartime destruction, natural
disasters, and strikes. For an open economy, shifts in terms of trade might
be a possible interpretation (for example due to temporary oil price shocks).
There is an alternative version of the RBC model which is based on the
specification = 1 and = 1 see Section 29.6.
4We recall that a sequence of stochastic variables with zero mean, constant variance,
and zero covariance across time is called white noise.5This version of the RBC model corresponds to that of the early RBC theorists, in-
cluding Prescott (1986) and King and Rebelo (1988). They adhered to the supposition
0 1 and had the cumulative aspect represented by a deterministic trend.
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29.1. A simple RBC model 1027
Factor demand
The representative firm is assumed to maximize its value under perfect com-
petition. Since there are no convex capital installation costs, the problem
reduces to that of static maximization of profits each period. And since pe-
riod ’s technological conditions ( , and the realization of) are assumed
known to the firm in period the firm does not face any uncertainty. Profit
maximization simply implies a standard factor demand ( ) satisfying
1( ) = + 0 ≤ ≤ 1 (29.4)
2() = (29.5)
where + is the real cost per unit of the capital service and is the real
wage.
The household
There is a given number of households or rather dynastic families, all alike
and with infinite horizon (Ramsey setup). For simplicity we ignore popula-
tion growth. Thus we consider a representative household of constant size
and with a constant amount of time at its disposal, say 1 time unit per pe-
riod. The household’s saving in period amounts to buying investment goods
that in the next period are rented out to the firms at the rental rate +1+ .
Thus the household obtains a net rate of return on financial wealth equal to
the interest rate +1
A decision problem under uncertainty
The preferences of the household are described by the expected discounted
utility hypothesis. Both consumption, and leisure, enter the period
utility function. Since the total time endowment of the household is one in
all periods, we have
+ = 1 = 0 1 2 (29.6)
where is labor supply in period The fact that has now been used
in two different meanings, in (29.1) as employment and in (29.6) as labor
supply, should not cause problems since in the competitive equilibrium of
the model the two are quantitatively the same.
The household has rational expectations. The decision problem, as seen
from time 0, is to choose current consumption, 0, and labor supply 0 as
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1028 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
well as a series of contingent plans, () and () for = 1 2 ,
so that expected discounted utility is maximized:
max0(0) = 0[
∞X=0
( 1−)(1 + )−] s.t. (29.7)
≥ 0 0 ≤ ≤ 1 (control region) (29.8)
+1 = (1 + ) + − 0 0 given (29.9)
+1 ≥ 0 for = 0 1 2 (29.10)
The period utility function (· ·) satisfies 1 0 2 0 11 0 22 0
and is concave.6 The decreasing marginal utility assumption implies, first, a
desire of smoothing over time both consumption and leisure; or we could say
that there is aversion towards variation over time in these entities. Second,
decreasing marginal utility reflects aversion towards variation in consumption
and leisure over different “states of nature”, i.e., risk aversion. The parameter
is the rate of time preference (the measure of impatience) and it is assumed
positive (a further restriction on will be introduced later).
When speaking of “period ” we mean the time interval [ + 1) The
symbol 0 signifies the expected value, conditional on the information avail-
able at the end of period 0. This information includes knowledge of all
variables up to period 0, including that period. There is uncertainty about
future values of and but the household knows the stochastic processes
which these variables follow (or, what amounts to the same, the stochastic
processes that lie behind, i.e., those for the productivity factor )
The constraint (29.9) displays our usual way of writing, in discrete time,
the dynamic accounting relation for wealth formation. An alternative way of
writing the condition is:
+1 − = + −
saying that private net saving is equal to income minus consumption. This
way of writing it corresponds to the form used in continuous time models.
We could also write
+1 − + = ( + ) + − (29.11)
saying that private gross investment is equal to gross income minus consump-
tion.
6Concavity implies adding the assumption 1122 − (12)2 ≥ 0
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29.1. A simple RBC model 1029
Characterizing the solution to the household’s problem
There are three endogenous variables, the control variables and and the
state variable The decision, as seen from period 0, is to choose a concrete
action (0 0) and a series of contingent plans (() ()) saying
what to do in each of the future periods as a function of the as yet unknown
circumstances, including the financial wealth, at that time. The decision
is made so that expected discounted utility 0(0) is maximized. The pair
of functions (() ()) is named a contingent plan because it refers
to what consumption and labor supply will be chosen in period in order to
maximize expected discounted utility, contingent on the financial wealth
at the beginning of period In turn this wealth, depends on the realized
path, up to period of the as yet unknown variable In order to choose
the action (0 0) in a rational way, the household must take into account
the whole future situation, including what the optimal contingent actions in
the future will be.
To be more specific, when deciding the action (0 0) the household
knows that in every new period it has to solve the remainder of the problem
as seen from that period. Defining ≡ (1 + ),7 the remainder of the
problem as seen from period ( = 0 1 ) is:
max = ( 1−) + (1 + )−1 [(+1 1−+1) (29.12)
+(+2 1−+2)(1 + )−1 + ¤
s.t. (29.8), (29.9), and (29.10), given.
To deal with this problem we will use the substitution method. First, from
(29.9) we have
= (1 + ) + −+1 and (29.13)
+1 = (1 + +1)+1 + +1+1 −+2 (29.14)
Substituting this into (29.12), the decision problem is reduced to an essen-
tially unconstrained maximization problem, namely one of maximizing the
function w.r.t. (+1) (+1+2) We first take the partial
derivative w.r.t. in (29.12) and set it equal to 0 (thus focusing on interior
solutions):
= 1( 1−) + 2( 1−)(−1) = 0
7Multiplying a utility function by a positive constant does not change the associated
optimal behavior.
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1030 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
which can be written
2( 1−) = 1( 1−) (29.15)
This first-order condition describes the trade-off between leisure in period
and consumption in the same period. The condition says that in the optimal
plan, the opportunity cost (in terms of foregone current utility) associated
with decreasing leisure by one unit equals the utility benefit of obtaining an
increased labor income and using this increase for extra consumption (i.e.,
marginal cost = marginal benefit, both measured in current utility).
Similarly, w.r.t. +1 we get the first-order condition8
+1
= 1( 1−)(−1) + (1 + )−1[1(+1 1−+1)(1 + +1)] = 0
This can be written
1( 1−) = (1 + )−1[1(+1 1−+1)(1 + +1)] (29.16)
where +1 is unknown in period This first-order condition describes the
trade-off between consumption in period and the uncertain consumption
in period + 1, as seen from period The optimal plan must satisfy that
the current utility loss associated with decreasing consumption by one unit
equals the discounted expected utility gain next period by having 1 + +1extra units available for consumption, namely the gross return on saving one
more unit (again, marginal cost = marginal benefit in utility terms). The
condition (29.16) is an example of a stochastic Euler equation. If there is
no uncertainty, the expectation operator can be deleted. Then, ignoring
the utility of leisure, (29.16) is the standard discrete-time analogue to the
Keynes-Ramsey rule in continuous time.
For completeness, let us also maximize explicitly w.r.t. the future pairs
(+++1) = 1 2 . We get
+
= (1 + )−1 [1(+ 1−+)+ + 2(+ 1−+)(−1)] = 0
so that
[2(+ 1−+)] = [1(+ 1−+)+]
8Generally speaking, for a given differentiable function (1 ) where is a
stochastic variable and 1 are parameters, we have
(( ))
=
( )
= 1
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29.1. A simple RBC model 1031
Similarly,
++1
=
£1(+ 1−+)(−1) + (1 + )−11(++1 1−++1)
·(1 + ++1) = 0
so that
[1(+ 1−+)] = (1 + )−1 [1(++1 1−++1)(1 + ++1)]
We see that for replaced by + 1 + 2 ... , (29.15) and (29.16) must hold
in expected values as seen from period The conclusion, so far, is that in
general, it suffices to write down (29.15) and (29.16) and then add that for
= 0 these two conditions are part of the set of first-order conditions and
for = 1 2 , similar first-order conditions hold in expected values.
Our first-order conditions say something about relative levels of consump-
tion and leisure in the same period and about the change in consumption
over time, not about the absolute levels of consumption and leisure. The
absolute levels are determined as the highest possible levels consistent with
the requirement that (29.15), (29.16), and (29.10), for = 0 1 2 ..., hold in
terms of expected values as seen from period 0. This can be shown to be
equivalent to requiring the transversality condition,
lim→∞
0£(1 + )−(−1)1(−1 1−−1)
¤= 0
satisfied in addition to the first-order conditions.9 Finding the resulting
consumption function requires specification of the period utility function.
But to solve for general equilibrium we do not need the consumption function.
As in a deterministic Ramsey model, knowledge of the first-order conditions
and the transversality condition is sufficient for determining the path over
time of the economy.
The remaining elements in the model
It only remains to consider the market clearing conditions. Implicitly we
have already assumed clearing in the factor markets, since we have used the
9In fact, in the budget constraint of the household’s optimization problem, we could
replace by financial wealth and allow borrowing, so that financial wealth could be neg-
ative. Then, instead of the non-negativity constraint (29.10), a No-Ponzi-Game condition
in expected value would be relevant. In a representative agent model with infinite horizon,
however, this does not change anything, since the non-negativity constraint (29.10) will
never be binding.
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1032 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
same symbol for capital and employment, respectively, in the firm’s problem
(the demand side) as in the household’s problem (the supply side). The
equilibrium factor prices are given by (29.4) and (29.5). We will rewrite
these two equations in a more convenient way. In view of constant returns
to scale, we have
= () = ( 1) ≡ () (29.17)
where ≡ () (the effective capital intensity). In terms of the in-
tensive production function (29.4) and (29.5) yield
+ = 1() = 0() (29.18)
= 2() =
h()−
0()i (29.19)
In a closed economy, by definition, gross investment ex post, , satisfies
+1 − = − (29.20)
Also, by definition, equals gross saving, − since, by simple expenditure
accounting,
= + (29.21)
Indeed, investment is in this model just the other side of households’ saving.
There is no independent investment function. To make sure that our national
expenditure accounting is consistent with our national income accounting
insert (29.20) into (29.21) to get
= +1 − + = − = ()− (29.22)
=
h
0() + ()− 0()
i−
= ( + ) + − (29.23)
where the second equality comes from (29.17) and the last equality follows
from (29.18) and (29.19). The result (29.23) is identical to the dynamic
budget constraint of the representative household, (29.11). Since this last
equation defines aggregate saving from the national income accounting, the
book-keeping is in order.
Specification of technology and preferences
To quantify the model we have to specify the production function and the
utility function. We abide by the common praxis in the RBC literature and
specify the production function to be Cobb-Douglas:
= 1− ()
0 1 (29.24)
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
29.2. A deterministic steady state 1033
Then we get
() = 1− (29.25)
+ = (1− )− (29.26)
= 1− (29.27)
As to the utility function we follow King and Rebelo (1999) and base the
analysis on the additively separable CRRA case,
( 1−) =1− − 11−
+ (1−)
1− − 11−
0 0 0
(29.28)
Here, is the (absolute) elasticity of marginal utility of consumption (the
desire for consumption smoothing), is the (absolute) elasticity of marginal
utility of leisure (the desire for leisure smoothing), and is the relative weight
given to leisure. In case or take the value 1, the corresponding term in
(29.28) should be replaced by log or log(1 − ) respectively. In fact,
most of the time King and Rebelo (1999) take both and to be 1.
With (29.28) applied to (29.15) and (29.16), we get
(1−)− = − and (29.29)
− =1
1 +
£−+1(1 + +1)
¤ (29.30)
respectively.
29.2 A deterministic steady state
For a while, let us ignore shocks. That is, assume = ∗ for all
The steady state solution
By a steady state we mean a path along which the growth-corrected variables
like and ≡ stay constant. With = ∗ for all (29.26) and(29.27) give the steady-state relations between and :
∗ =
∙(1− )∗
∗ +
¸1 (29.31)
∗ = ∗∗1− (29.32)
We may write (29.30) as
1 + =
∙(+1
)−(1 + +1)
¸ (29.33)
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1034 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
In the non-stochastic steady state the expectation operator can be deleted,
and and are independent of Hence, +1 = by (29.2), and
(29.33) takes the form
1 + ∗ = (1 + ) (29.34)
In this expression we recognize the modified golden rule discussed in chapters
7 and 10.10 Existence of general equilibrium in our Ramsey framework re-
quires that the long-run real interest rate is larger than the long-run output
growth rate, i.e., we need ∗ − 1 This condition is satisfied if and only if
1 + 1− (29.35)
which we assume.11 If we guess that = 1 and = 001 then with = 1004
(taken from US national income accounting data 1947-96, using a quarter
of a year as our time unit), we find the steady-state rate of return to be ∗
= 0014 or 0056 per annum. Or, the other way round, observing the average
return on the Standard & Poor 500 Index over the same period to be 6.5 per
annum, given = 1 and = 1004 we estimate to be 0012
Using that in steady state is a constant, ∗ we can write (29.22) as
+1 − (1− ) = ∗1− − (29.36)
where ≡ (∗) Given ∗ (29.31) yields the steady-state capital in-
tensity ∗ Then, (29.36) gives
∗ ≡ ∗
= ∗∗1− − ( + − 1)∗
Consumption dynamics around the steady state in case of no un-
certainty
The adjustment process for consumption, absent uncertainty, is given by
(29.33) as
(+1
)−(1 + +1) = 1 +
or, taking logs,
log+1
=1
[log(1 + +1)− log(1 + )] (29.37)
10King and Rebelo, 1999, p. 947, express this in terms of the growth-adjusted discount
factor ≡ (1 + )−11− so that 1 + ∗ = (1 + ) = 11Since 1 only if 1 (which does not seem realistic, cf. Chapter 3), is it possible
that 0 is not sufficient for (29.35) to be satisfied.
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
29.3. On the approximate solution and numerical simulation 1035
This is the deterministic Keynes-Ramsey rule in discrete time with separable
CRRA utility. For any “small” we have log(1 + ) ≈ (from a first-order
Taylor approximation of log(1 + ) around 0) Hence, with = +1− 1we have log(+1) ≈ +1−1 so that (29.37) implies the approximaterelation
+1 −
≈ 1(+1 − ) (29.38)
There is a supplementary way of writing the Keynes-Ramsey rule. Note
that (29.34) implies log(1+∗) = log(1+)+ log Using first-order Taylor
approximations, this gives ∗ ≈ + log ≈ + where ≡ − 1 is thetrend rate of technological progress. Thus ≈ ∗ − and inserting this
into (29.38) we get
+1 −
≈ 1(+1 − ∗) +
Then the technology-corrected consumption level, ≡ moves accord-
ing to+1 −
≈ 1
(+1 − ∗)
since is the growth rate of
29.3 On the approximate solution and nu-
merical simulation
In the special case = 1 (the log utility case), still maintaining the Cobb-
Douglas specification of the production function, the model can be solved
analytically provided capital is non-durable (i.e., = 1).12 It turns out that
in this case the solution has consumption as a constant fraction of output (i.e.,
there is a constant saving rate as in the Solow growth model). Further, in this
special case labor supply equals a constant and is thus independent of the
productivity shocks. Since in actual business cycles, employment fluctuates
a lot, this might not seem to be good news for a business cycle model.
But assuming = 1 for a period length of one quarter or one year is “far
out”. Given a period length of one year, is generally estimated to be less
than 0.1. And with 1 labor supply is affected by the technology shocks.
An exact analytical solution, however, can no longer be found.
12Alternatively, even allowing 1 one could coarsely assume that it is “gross-gross
output”, i.e., GDP + (1− ), that is described by a Cobb-Douglas production function.
Then, the model could again be solved analytically.
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1036 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
One can find an approximate solution based on a log-linearization of the
model around the steady state. Without dwelling on the more technical
details we will make a few observations.
29.3.1 Log-linearization
If ∗ is the steady-state value of the variable in the non-stochastic case,then one defines the new variable
≡ log(∗) = log − log ∗ (29.39)
That is, is the logarithmic deviation of from its steady-state value. But
this is approximately the same as ’s proportionate deviation from its steady-
state value. This is because, when is in a neighborhood of its steady-state
value, a first-order Taylor approximation of log around ∗ gives
log ≈ log ∗ + 1
∗( − ∗)
so that
≈ − ∗
∗ (29.40)
Working with the transformation instead of implies the convenience
that
+1 − = log(+1
∗)− log(
∗) = log +1 − log
≈ +1 −
That is, relative changes in have been replaced by absolute changes in
Some of the equations of interest are exactly log-linear from start. This
is true for the production conditions (29.25), (29.26), and (29.27) as well as
for the first-order condition (29.29) for the household. For other equations
log-linearization requires approximation. Consider for example the time con-
straint (29.6). With denoting leisure (≡ 1−) this constraint implies
∗ −∗
∗ + ∗ − ∗
∗= 0
or
∗ + ∗ ≈ 0 (29.41)
by the principle in (29.40). From (29.29), taking into account that 1− =
, we have
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
29.3. On the approximate solution and numerical simulation 1037
− = − ≡ ()
−
= −
1− (29.42)
In steady state this takes the form
∗− = ∗−∗1− (29.43)
We see that when 1 (sustained technological progress), we need = 1
for a steady state to exist (which explains why in their calibration King and
Rebelo assume = 1). This quite “narrow” theoretical requirement is an
unwelcome feature and is due to the additively separable utility function
assumed by King and Rebelo.
Combining (29.43) with (29.42) givesµ
∗
¶−=³ ∗
´−
∗
Taking logs on both sides we get
− log
∗= log
∗− log
∗
or
− = −
In view of (29.41), this implies
= − ∗
∗ =1−∗
∗ − 1−∗
∗ (29.44)
This result tells us that the elasticity of labor supply w.r.t. a tempo-
rary change in the real wage depends negatively on 13 Indeed, calling this
elasticity , we have
=1−∗
∗ (29.45)
Departing from the steady state, a one per cent increase in the wage ( =
001) leads to an per cent increase in the labor supply, by (29.44) and
(29.45). The number measures a kind of compensated wage elasticity of
labor supply (in an intertemporal setting), relevant for evaluating the pure
substitution effect of a temporary rise in the wage. King and Rebelo (1999)
reckon ∗ in the US to be 0.2 (that is, out of available time one fifth is
13This is not surprising, since reflects the desire for leisure smoothing across time.
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1038 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
working time). With = 1 (as in most of the simulations run by King and
Rebelo), we then get = 4 This elasticity is much higher than what the
micro-econometric evidence suggests (at least for men), namely typically an
elasticity below 1 (Pencavel 1986). But with labor supply elasticity as low as
1, the RBCmodel is far from capable of generating a volatility in employment
comparable to what the data show.
For some purposes it is convenient to have the endogenous time-dependent
variables appearing separately in the stationary dynamic system. Then, to
describe the supply of output in log-linear form, let ≡ ≡ ()
and ≡ ≡ From (29.24),
= 1−
and dividing through by the corresponding expression in steady state, we get
∗=
∗(
∗)1−(
∗ )
Taking logs on both sides we end up with
= + (1− ) + (29.46)
For the demand side we can obtain at least an approximate log-linear
relation. Indeed, dividing trough by in (29.21) we get
+ =
where ≡ Dividing through by ∗ and reordering, this can also bewritten
∗
∗ − ∗
∗+
∗
∗ − ∗
∗=
− ∗
∗
which, using the hat notation from (29.40), can be written
∗
∗ +
∗
∗ ≈ (29.47)
to be equated with the right hand side of (29.46).
29.3.2 Numerical simulation
After log-linearization the model can be reduced to two coupled linear sto-
chastic first-order difference equations in and where is predetermined,
and is a jump variable. There are different methods available for solving
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
29.3. On the approximate solution and numerical simulation 1039
such an approximate dynamic system analytically.14 Alternatively, based on
a specified set of parameter values one can solve the system by numerical
simulation on a computer.
In any case, when it comes to checking the quantitative performance of
the model, RBC theorists generally stick to calibration, that is, the method
based on a choice of parameter values such that the model matches a list of
data characteristics. In the present context this means that:
(a) the structural parameters ( ∗) are given values thatare taken or constructed partly from national income accounting and
similar data, partly from micro-econometric studies of households’ and
firms’ behavior;
(b) the values of the parameters, and in the stochastic process for
the productivity variable are chosen either on the basis of data for
the Solow residual15 over a long time period, or one or both values are
chosen to yield, as closely as possible, a correspondence between the sta-
tistical moments (standard deviation, auto-correlation etc.) predicted
by the model and those in the data.
The first approach to and is followed by, e.g., Prescott (1986). It
has been severely criticized by, among others, Mankiw (1989). In the short
and medium term, the Solow residual is very sensitive to labor hoarding and
variations in the degree of utilization of capital equipment. It can therefore
be argued that it is the business cycle fluctuations that explain the fluctua-
tions in the Solow residual, rather than the other way round.16 The second
approach, used by, e.g., Hansen (1985) and Plosser (1989), has the disadvan-
tage that it provides no independent information on the stochastic process
for productivity shocks. Yet such information is necessary to assess whether
14For details one may consult Campbell (1994, p. 468 ff.), Obstfeld and Rogoff (1996,
p. 503 ff.), or Uhlig (1999).15Given (29.24), take logs on both sides to get
log = log + (1− ) log + log + log
Then the Solow residual; may be defined by
log ≡ log + log = log − (1− ) log − log
16King and Rebelo (1999, p. 982-993) believe that the problem can be overcome by
refinement of the RBC model.
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1040 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
the shocks can be the driving force behind business cycles.17
As hitherto we abide to the approach of King and Rebelo (1999) which
like Prescott’s is based on the Solow residual. The parameters chosen are
shown in Table 19.2. Remember that the time unit is a quarter of a year.
Table 29.2. Parameter values
∗ 0.667 0.025 0.0163 1 1 3.48 1.004 0.2 0.979 0.0072
Given these parameter values and initial values of and in confor-
mity with the steady state, the simulation is ready to be started. The shock
process is activated and the resulting evolution of the endogenous variables
generated through the “propagation mechanism” of the model calculated by
the computer. From this evolution the analyst next calculates the different
relevant statistics: standard deviation (as an indicator of volatility), auto-
correlation (as an indicator of degree of persistence), and cross correlations
with different leads and lags (reflecting the co-movements and dynamic inter-
action of the different variables). These model-generated statistics can then
be compared to those calculated on the basis of the empirical observations.
In order to visualize the economic mechanisms involved, impulse-response
functions are calculated. Shocks before period 0 are ignored and the economy
is assumed to be in steady state until this period. Then, a positive once-for-
all shock to occurs so that productivity is increased by, say, 1 % (i.e., given
−1 = ∗ = 1 we put 0 = 0 01 in (29.3) with = 0). The resulting path forthe endogenous variables is calculated under the assumption that no further
shocks occur (i.e., = 0 for = 1 2 ) An “inpulse-response diagram”
shows the implied time profiles for the different variables.
Remark. The text should here show some graphs of impulse-response
functions. These graphs are not yet available. Instead the reader is referred
to the graphs in King and Rebelo (1999), p. 966-970. As expected, the
time profiles for output, consumption, employment, real wages, and other
variables differ, depending on the size of in (29.3). Comparing the case
17At any rate, calibration is different from econometric estimation and testing in the
formal sense. Criteria for what constitutes a good fit are not offered. The calibration
method should rather be seen as a first check whether the model is logically capable of
matching main features of the data (say the first and second moments of key variables).
Calibration delivers a quantitative example of the working of the model. It does not deliver
an econometric test of the validity of the model or of a hypothesis based on the model.
Neither does it provide any formal guide as to what aspects of the model should be revised
(see Hoover, 1995, pp. 24-44).
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
29.4. The two basic propagation mechanisms 1041
= 0 (a purely temporary productivity shock) and the case = 0979 (a
highly persistent productivity shock), we see that the responses are more
drawn out over time in the latter case. This persistence in the endogenous
variables is, however, just inherited from the assumed persistence in the
shock. And amplification is limited. When is high, in particular when
= 1 (a permanent productivity shock), wealth effects on labor supply are
strong and dampen the substitution effect.
29.4 The two basic propagation mechanisms
We have added technology shocks to a standard neoclassical growth model
(utility-maximizing households, profit-maximizing firms, rational expecta-
tions, market clearing under perfect competition). The conclusion is that
correlated fluctuations in output, consumption, investment, work hours, out-
put per man-hour, real wages, and the real interest rate are generated. So
far so good. There are two basic “propagation” mechanisms (transmission
mechanisms) that drive the fluctuations:
1. The capital accumulation mechanism. To understand this mechanism
in its pure form, let us abstract from the endogenous labor supply and
assume an inelastic labor supply. A positive productivity shock in-
creases marginal productivity of capital and labor. If the shock is not
purely temporary, the household feels more wealthy. Both output, con-
sumption and saving (due to intertemporal substitution in consump-
tion) go up. The increased capital stock implies higher output also in
the next periods. Hence output shows positive auto-correlation (per-
sistence). And output, consumption, and investment move together
(co-movement).
2. Intertemporal substitution in labor supply. An immediate implication of
increased marginal productivity of labor is a higher real wage. To the
extent that this increased real wage is only temporary, the household
is motivated to supply more labor in the current period and less later.
This is the phenomenon of intertemporal substitution in leisure. By the
adherents of the RBC theory the observed fluctuations in work hours
are seen as reflecting this.
29.5 Limitations
During the last 15-20 years there has been an increasing scepticism towards
the RBC theory. The main limitation of the theory is seen to derive from
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1042 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
its insistence upon interpreting fluctuations in employment as reflecting fluc-
tuations in labor supply. The critics maintain that, starting from market
clearing based on flexible prices, it is not surprising that difficulties match-
ing the business cycle facts arise.
We may summarize the objections to the theory in the following four
points:
a. Where are the productivity shocks? As some critics ask: “If produc-
tivity shocks are so important, why don’t we read about them in the
Wall Street Journal?” For example, it definitely seems hard to interpret
the absolute economic contractions (decreases in GDP) that sometimes
occur in the real world as due to productivity shocks. If the elasticity
of output w.r.t. productivity shocks does not exceed one (as it does
not seem to, empirically, according to Campbell 1994), then a backward
step in technology at the aggregate level is needed. Although sound
technological knowledge as such is always increasing, mistakes could
be made in choosing technologies. At the disaggregate level, one can
sometimes identify technological mistakes, like the use of DDT and its
subsequent ban in the 1960’s due to its damaging effects on health.
But it is very hard to think of technological drawbacks at the aggregate
level, capable of explaining the observed economic recessions. Think
of the large and long-lasting contraction of GDP in the US during the
Great Depression (27 % reduction between 1929 and 1933 according to
Romer, 2001, p. 171). Sometimes the adherents of the RBC theory
refer also to other kinds of supply shocks: changes in taxation, changes
in environmental legislation etc. (Hansen and Prescott, 1993). But the
problem is that significant changes in taxation and regulation occur
rather infrequently and are therefore not a convincing candidate for
the driving force in the stochastic process (29.3).
b. Lack of internal propagation. Given the micro-econometric evidence
that we have, the two mechanisms above seem far from capable at gen-
erating the large fluctuations that we observe. Both mechanisms imply
too little “amplification” of the shocks. Intertemporal substitution in
labor supply does not seem able of generating much amplification. This
is related to the fact that changes in real wages tend to be permanent
rather than purely transitory. Permanent wage increases tend to have
little or no effect on labor supply (the wealth effect tends to offset
the substitution and income effect). Given the very minor temporary
movements in the real wage that occur at the empirical level, a high
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
29.6. Technological change as a random walk with drift 1043
intertemporal elasticity of substitution in labor supply18 is required to
generate the large fluctuations in employment observed in the data.
But the empirical evidence suggests that this requirement is not met.
Micro-econometric studies of labor supply indicate that this elasticity,
at least for men, is quite small (in the range 0 to 1.5, typically below
1).19 Yet, Prescott (1986) and Plosser (1989) assume it is around 4.
c. Correlation puzzles. Sometimes the sign, sometimes the size of cor-
relation coefficients seem persevering wrong (see King and Rebelo, p.
957, 961). As Akerlof (2003, p. 414) points out, there is a conflict be-
tween the empirically observed pro-cyclical behavior of workers’ quits20
and the theory’s prediction that quits should increase in cyclical down-
turns (since variation in employment is voluntary according to the the-
ory). Considering a dozen of OECD countries, Danthine and Donaldson
(1993) find that the required positive correlation between labor produc-
tivity and output is visible only in data for the U.S. (and not strong),
whereas the correlation is markedly negative for the majority of the
other countries.
d. Disregard of non-neutrality of money. According to many critics, the
RBC theory conflicts with the empirical evidence of the real effects of
monetary policy.
Numerous, and more and more imaginative, attempts at overcoming the
criticisms have been made; King and Rebelo (1999, p. 974-993) present some
of these. In particular, adherents of the RBC theory have looked for mech-
anisms that may raise the size of labor supply elasticities at the aggregate
level over and above that at the individual level found in microeconometric
studies.
29.6 Technological change as a random walk
with drift
In contrast to Prescott (1986) and King and Rebelo (1999), Plosser (1989)
assumes that technological change is a random walk with drift. The repre-
18Recall that this is defined as the percentage increase (calculated along a given indif-
ference curve) in the ratio of labor supplies in two succeeding periods prompted by a one
percentage increase in the corresponding wage ratio, cf. Chapter 5.19Handbook of Labor Economics, vol. 1, 1986, Table 1.22, last column. See also Hall
(1999, p. 1148 ff.).20See Chapter 28.
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
1044 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
sentative firm has the production function
= ( )
where is a measure of the level of technology, and the production function
has constant returns to scale. In the numerical simulation Plosser used a
Cobb-Douglas function.
The technology variable (total factor productivity) is an exogenous
stochastic variable. In contrast to the process for the logarithm of above,
where we had 1 we now assume a “unit root”, i.e., = 1 So the process
assumed for ≡ log is
= + −1 + (29.48)
a random walk. This corresponds to our general conception of technical
knowledge as cumulative. If the deterministic term 6= 0, the process is
called a random walk with drift. In the present setting we can interpret
as some underlying given trend in productivity, suggesting 021 Negative
occurrences of the noise term need not in this case represent “technological
regress”, but just a technology development below trend (which will be the
case if − ≤ 0)
This version of the RBCmodel also faces difficulties. Indeed, embedded in
aWalrasian equilibrium framework the specification (29.48) tends to generate
too little fluctuation in employment and output. This is because, when shocks
are permanent, large wealth effects offset the intertemporal substitution in
labor supply.
29.7 Conclusion
It is advisory to make a distinction between on the one hand RBC theory
(based on perfect competition and market clearing in an environment where
productivity shocks are the driving force behind the fluctuations) and on the
other hand the quantitative modeling framework known as DSGE models.
A significant amount of research on business cycle fluctuations has left the
RBC theory, but applies similar quantitative methods. This approach consists
in an attempt at building small quantitative Dynamic Stochastic General
21The growth rate in total factor productivity is ( − −1) −1. From (29.48) we
have −1 ( − −1) = , and − −1 = log − log−1 ≈ ( − −1) −1 by a1. order Taylor approximation of log about −1 Hence, −1 ( − −1) −1 ≈
In Plosser’s model all technological change is represented by change in i.e., in (29.2)
Plosser has = 1
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
29.8. Literature notes 1045
Equilibrium models. The economic content of such a model can be New
Classical (as with Lucas and Prescott). Alternatively it can be more or
less New Keynesian, based on a combination of imperfect competition and
other market imperfections (also in the financial markets), and nominal and
real price rigidities (see, e.g., Jeanne, 1998, Smets and Wouters, 2003, and
Danthine and Kurmann, 2004, Gali, 2008).
Medium-term theory attempts to throw light on business cycle fluctua-
tions and to clarify what kinds of counter-cyclical economic policy, if any, may
be functional. This is probably the area within macroeconomics where there
is most disagreement − and has been so for a long time. Some illustratingquotations:
Indeed, if the economy did not display the business cycle phenom-
ena, there would be a puzzle. ... costly efforts at stabilization are
likely to be counterproductive. Economic fluctuations are opti-
mal responses to uncertainty in the rate of technological change
(Prescott 1986).
My view is that real business cycle models of the type urged on us
by Prescott have nothing to do with the business cycle phenomena
observed in the United States or other capitalist economies. ...
The image of a big loose tent flapping in the wind comes to mind
(Summers 1986).
29.8 Literature notes
In dealing with the intertemporal decision problem of the household we ap-
plied the substitution method. More advanced approaches include the dis-
crete time Maximum Principle (see Chapter 8), the Lagrange method (see,
e.g., King and Rebelo, 1999), or Dynamic Programming (see, e.g., Ljungqvist
and Sargent, 2004).
29.9 Exercises
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1046 CHAPTER 29. THE REAL BUSINESS CYCLE THEORY
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
Chapter 30
Keynesian perspectives on
business cycles
Applying a Vector-Autoregression time series approach with two kinds of
shocks interpreted as demand and supply shocks, respectively, Blanchard and
Quah (1989) found on the basis of quarterly US data 1950-87 that demand
shocks explain more that two thirds of the fluctuations in output and even
more of the fluctuations in unemployment. Working with a somewhat larger
system and quarterly US data for 1965-1986, Blanchard (1989) summarized
the results this way:
(a) Demand shocks explain most of the short-run fluctuations in output.
(b) Positive demand shocks are associated with gradual increases in nomi-
nal prices and wages.
(c) Supply shocks dominate the medium and the long run, and positive
supply shocks are associated with decreases in nominal prices and wages
(relative to trend).
This leads to a Keynesian understanding of macroeconomic fluctuations.
Point (a) indicates that demand shocks, such as a shift in the state of con-
fidence, a shift in government spending, a shift in liquidity preference, or a
sudden tightening of credit to begin with have a larger effect on output than
on prices. The interpretation is that nominal and relative price rigidities lie
underneath. Then, point (b) reminds us that even though prices in the ma-
jor sectors of the economy react only sluggishly to demand shocks, they do
react over time via cost push due to changes in the level of economic activity
(the Phillips curve). Finally, point (c) says that durable influences on output
1047
1048
CHAPTER 30. KEYNESIAN PERSPECTIVES ON BUSINESS
CYCLES
come from supply factors, such as the labor force, capital, and technological
change.
Characteristic of the Keynesian understanding of economic fluctuations
is the emphasis on the sometimes vicious, sometimes virtuous circles that
in particular may arise when production is demand-determined rather than
supply-determined. In continuation of the emphasis on nominal price sticki-
ness, a crucial element in the Keynesian approach is the refutation of Say’s
law. This is the “law” claiming that “supply creates its own demand” or
that “income is automatically spent on produced goods and services”. As
described in Chapter 19, Keynes’ refutation of this doctrine rests on a re-
jection of the validity of the Walrasian budget constraint and Walrasian de-
mands and supplies when trade occurs outside Walrasian equilibrium. In the
Keynesian approach these notions are replaced by the concepts of effective
budget constraints and effective demands and supplies.
In recent decades there has been a tendency to downplay the differences
between Keynesian and new-Classical thinking. In certain respects a kind
of theoretical convergence has emerged in the sense that many of the earlier
new-Classicals now explore the possible role of nominal rigidities and other
market imperfections. And new-Keynesians attempt to avoid some of the
shortcomings of old Keynesian theory pointed out by the new-classicals and
others. Nevertheless, in the wake of the Great Recession (2008-) we have
witnessed a fresh outbreak of disagreements between Keynesians and new-
Classicals, especially in USA. This comes into view both at the theoretical
level and in regard to opinions about economic policy.
We use the label “Keynesian” when relevant to bring to light differences
vis-a-vis classical (or Walrasian) macroeconomic thinking. Although which is
utterly interwoven with Say’s law. This chapter aims at a broad description
of the elements in the Keynesian understanding of business cycles.1 Our
account begins with the short run and the associated building blocks.
30.1 The short run
The rule of the minimum
In both goods and labor markets dominated by imperfect competition, prices
and wages are set by agents with market power. Owing to menu costs, prices
and wages are sticky in the short run and quantities supplied are determined
by the rule of the minimum. In general this rule makes effective demand the
binding constraint. Given the preset , firm ’s ex post production level
1A more specific new-Keynesian “workhorse model” is described in Chapter 32.
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30.1. The short run 1049
becomes
= min£( )
( )¤ (30.1)
where is the general price level, is aggregate demand, the general
nominal wage level, ( ) demand faced by firm and ( ) the
classical supply at the preset price and the going wage. In Chapter 19 this
principle, called the rule of the minimum, is described in detail. Demand,
( ) will generally define the most narrow limit and so production
will be determined by demand. Thus, the firm will be producing a quantity
which is sometimes below the level where = sometimes above,
but usually below the level where = Under “normal circumstances”
the whole setup makes room for considerable variation in employment and
output without any price and wage change in the short run.
The components of aggregate demand
In the modeling of the components of aggregate demand, some models ap-
ply the device of a representative household, an approach towards which
other economists are sceptical when it comes to understanding business cy-
cles. A shared feature across different Keynesians is, however, the emphasis
on market failures that restrict the opportunities faced by the agents. In
the consumption function typically not only wealth and the (long-term) real
interest rate will enter, but also current income, because of the presump-
tion that many households are credit constrained. In this way credit market
imperfections are an additional channel, besides involuntary unemployment,
through which not only price signals, but also quantity signals become de-
terminants of consumption demand.
Fixed capital investment is typically modelled in a way akin to the -
theory of investment. But compared to the presentation of the -theory in
Chapter 14, where the firm’s environment was competitive, under imperfect
competition also expected future demand, that is, a perceived quantity signal,
becomes important for marginal
Net exports are typically modelled as an increasing function of the real
exchange rate (degree of “competitiveness”) and a decreasing function of
as in Chapter 21.
Asset markets
In the traditional Keynesian IS-LM model there are only two assets, money
and standardized short-term bonds bearing a nominal interest rate, , and
traded in a centralized auction market. Real money demand is then given by
the simple money demand function = ( ) where is the general
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price level, sticky in the short run, and 0 and 0, reflecting the
transaction motive for holding money and the opportunity cost of holding
money, respectively. The money supply, , is either the monetary base or a
multiple of the monetary base assumed under control by the central bank.
In the market for produced goods it may, in a short-run perspective,
take time for production to adjust to demand, but the asset markets are
fast-moving and clear instantaneously by adjustment of the interest rate,
or the money supply, depending on the monetary policy regime. Thus,
= ( ) at any point in time. In view of the balance sheet constraint,
equilibrium in the money market implies equilibrium in the bond market
and vice versa. In this setup, changes in the money supply affects aggregate
demand via changes in which in turn affects the expected real interest rate
= − where is the expected inflation rate. We say that the monetarytransmission goes via the interest rate channel.
In the late 1980s also the bank lending channel began to be emphasized
(Bernanke and Blinder, 1988). The viewpoint is that customer bank loans
are in many countries the main source of finance for small firms and house-
holds. At the same time bank loans play a critical role during times of
financial stress. In the formal models there are thus two types of loans,
standardized relatively safe bonds and personalized risky bank loans, with
corresponding interest rates, and The commercial banks are likely
to economize more with respect to their excess reserves (in excess over the
required reserves), the higher are both the bond rate and the bank lending
rate. Thereby, the money multiplier depends positively on these rates.
More advanced models also consider the role of the stock market and the
housing market for the macroeconomy. The market value of these assets is
important for households’ consumption and firms’ investment.
30.2 From the short to the medium run
To go from the short to the medium run requires bringing dynamics into
the picture. Dynamics is about how the state of the economy in the current
period gives rise to actions leading to a certain state in the next period and
so on. The links between the periods come from several factors.
The emphasis on aggregate demand does not mean that the supply side
of the economy is regarded as unimportant. The supply side embodies the
dominant market form, imperfect competition with price setting agents. The
supply side determines a ceiling on output. And the supply side is also
important for changes in expectations and the rate of inflation as manifested
in the Phillips curve. In turn, the Phillips curve is an essential factor in the
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30.2. From the short to the medium run 1051
short-to-medium-run dynamics of the economy.
30.2.1 Changes in expectations
The Keynesian modeling of the short run usually take agents’ expectations to
be essentially exogenous (“temporary equilibria”). In a medium-run perspec-
tive changes in expectations become important. But expectation formation
is a difficult field and far from settled. At least the old-Keynesian approach
is not interlocked with any specific hypothesis about expectation formation.
Depending on the circumstances, alternative hypotheses are introduced, in-
cluding rational expectations, adaptive expectations, extrapolative expecta-
tions, and adaptive learning - as well as mixtures of these, thus allowing
heterogeneity across the agents.
There is among theorists an increasing recognition that models should
incorporate that economic agents quite often disagree in their conceptions
about how the economy functions and therefore disagree in their expectations
about the future. A large part of the trade in the asset markets in fact seems
induced by such disagreements.
The “majority opinion” among the market participants may shift, some-
times slowly, sometimes fast (information cascades). Example: for a decade
up to 2006/07 house prices in many countries were rising much faster than
the economy and faster than construction costs. House prices seemed sys-
tematically diverging from the fundamental value. In such a situation some
people begin to recognize that the evolving explosive price path in the hous-
ing market is probably not sustainable. But at the same time many market
participants feel a deep-seated uncertainty about when the tipping point will
arrive - simply because one cannot know. Hence, it may not be irrational to
continue speculation in further price rises for some time. The turning point
will not arrive until there are enough market participants who believe that
many of the other market participants believe that many of the other ... etc.
are now changing their optimistic view.
30.2.2 Phillips curve/wage curve
Many Keynesian models linking the short to the medium run employ some
kind of an expectations-augmented Phillips curve. There are alternative ways
of modelling the details in this relation which involves both the labor market
and the goods market. Here we just give a broad picture.
Macroeconometric evidence indicates, in particular for the US after the
Second World War, a negative relation between the rate of change of wages
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and the unemployment rate:
− −1 = + (−1 − −2)− + (30.2)
= + (−1 − −2)− ( − ) +
where = ln , = ln , is the unemployment rate, and and are
positive constants, whereas is an error term.2 This is a wage Phillips
curve. One interpretation is this. As appears in the second line of (30.2),
the parameter can be split into a sum of two terms, which indicates
the long-run growth rate in labor productivity, and a term where
≡ (−) (to be interpreted below as the “natural” rate of unemployment).A straightforward reading of the role of the (lagged) inflation term, −1−−2in (30.2) is that it represents expected inflation. Let denote the expected
price level in period as seen from the end of period − 1 and let denotethe expected inflation rate, i.e., ≡ (
− −1)−1 ≈ − −1 Then,according to the hypothesis of static expectations of the inflation rate we
have
− −1 = −1 − −2 (30.3)
In fact, if inflation follows a random walk (which the data does not reject3),
this hypothesis is consistent with rational expectations.
Substituting (30.3) into (30.2) and ordering gives the expected change of
the real wage as a decreasing function of unemployment:
− − (−1 − −1) = − ( − ) + (30.4)
In this way the empirical Wage Phillips curve, (30.2), is seen as reflecting
an expected-real-wage Phillips curve. If expectations are not systematically
wrong and the trend rate of unemployment is close to this says that real
wages tend in the long run to grow at the same rate as labor productivity,
The data for the US roughly confirms this picture. Consequently, a first
interpretation of is that it is that rate of unemployment which is consistent
with real wages tending to grow at the same rate as labor productivity.
Whatever the interpretation of (30.2), it can under a certain condition
be transformed into a price Phillips curve. Suppose prices are formed by a
more or less constant mark-up on marginal cost, = (1 + ) where
is labor productivity. Then roughly the price inflation rate equals the
wage inflation rate minus the productivity growth rate,
− −1 = − −1 −
2See, e.g., Blanchard and Katz (1999).3See Hendry (2008).
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30.2. From the short to the medium run 1053
Substituting this into (30.2) gives
− −1 = −1 − −2 − ( − ) + (30.5)
Thus, if inflation increases, and if inflation decreases. This
corresponds to the interpretation of as the NAIRU (Non-Accelerating-
Inflation-Rate of Unemployment) in the sense of that rate of unemployment
which is consistent with a constant inflation rate; is also sometimes called
the “natural” or the “structural” rate of unemployment.
As discussed by Blanchard and Katz (1999), the Phillips curve (30.2) fits
European data less well than US data. And at the theoretical level it is in
fact not obvious why a Phillips curve should hold in the first place. According
to the theories of the functioning of labor markets (efficiency wages, social
norms, search theories, and bargaining) it is the level of the expected real
wage, rather than the expected change in the real wage, that is negatively
related to unemployment. Theory thus predicts a wage curve:
− = + (1− ) − + (30.6)
where is a constant ∈ [0 1] is the reservation wage (the minimum real
wage at which the worker is willing to supply labor), and a measure of
labor productivity.
By reasonable hypotheses about how the reservation wage depends on
the actual real wage (in the previous period) and on productivity, a level
formulation as in (30.6) may be consistent with a change formulation as
in (30.2). Blanchard and Katz (1999) find such consistency to be plausible
for US labor markets, but not for the typical European labor market with
more influential labor unions, more stringent hiring and firing regulations,
and perhaps also a greater role of the underground economy. An interesting
implication of this theory is that in Europe, the NAIRU should be sensitive
to permanent shifts in factors such as the level of energy prices, payroll taxes,
or real interest rates, whereas in the US it should not.
30.2.3 Other dynamic links
Net investment in fixed capital in the current period leads to more productive
capacity in the next period. Thereby the classical supply, in the produc-
tion lines involved is raised, cf. (30.1), thus providing more scope for upward
adjustment of output when demand goes up. Another link has to do with
inventory investment. After a sudden upturn of the economy, for example,
there will be a need to replenish inventories. So firms will for a while produce
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at a rate above what is needed to satisfy the demand by final users. This
reinforces the upturn.4
Also fiscal and monetary policies imply links from one period to the next.
Because of the presumed existence of nominal rigidities, monetary policy
rules have received much attention in recent research. For several countries
with floating exchange rate there is ample empirical basis for claiming that
their monetary policy follows a Taylor rule, a reaction function of the form:
= max [0 0 + 1( − ∗ ) + 2( − )] 1 ≥ 0 2 1 (30.7)
where ∗ is the log of “natural” (or trend) level of output, the actualinflation rate (sometimes replaced by the expected inflation rate), and
the inflation target; 0 + − ≡ indicates the implicit real interest
rate faced by the ultimate borrowers when = ∗ and = including
the interest differential ≥ 0 reflecting risk. The Taylor rule thus gives thetarget nominal interest rate chosen by the central bank as an increasing linear
function of the output gap, − ∗ and the excess inflation rate, − ; the
rule has 2 1 to make the target real interest rate an increasing function of
the actual inflation rate, thereby achieving a stabilizing role. In some versions
of the Taylor rule it is rather the expected inflation rate and, sometimes, the
expected output level in the near future that enter the reaction function.
30.2.4 Aren’t desirable adjustments automatic and fast?
Even positive supply shocks may at first lead to lower employment. Time
series evidence for this has been provided by for instance Gali (1999). This
observation is at odds with the real business cycle theory but fits well with the
Keynesian understanding of business cycles. To see this, let us return to the
rule of the minimum in Section 30.1, suppose organizational or technological
improvements increase the productivity level. As long as the wage remains
unchanged, this implies that marginal cost for any given output level tends
to become lower. But as long as the general demand level, has not risen
and and are unaltered, then will also remain unaltered. Consequently
the firm needs less labor than before. Then we get lower employment, lower
labor income, and thereby, in fact, lower aggregate demand. The point is
that structural improvements need not immediately get things going.
Two factors which operate in the positive direction, though, are, first,
that the reduction in costs tends to lower prices and increase real wages over
4The self-fulfilling prophesy investment theory by Kiyotaki (1988) and the inventory
investment theory by Blinder ( ) are examples of business cycle theory emphasizing
firms’ investment.
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30.2. From the short to the medium run 1055
time and can thereby increase real demand and income. The expectation of
higher real wages in the future makes people feel wealthier and this tends to
increase consumption already now. Second, when new technologies furnish
society with new consumption goods, demand tends to be stimulated (think
of the ICT revolution).
Other aspects of expectations will be involved, however, and they add to
the complexity of the situation. Falling prices do not always ensure greater
income. This is because demand can respond negatively. An expectation
of further decreases in prices can lead to deferment of purchases of durable
consumption and investment goods. Or stated differently: for a given nomi-
nal interest rate, expected deflation implies a greater real interest rate than
if prices were constant. In this way an economic downturn which leads to
deflation tends thereby to reinforce itself. The Great Depression in the US
in the 1930s and the long period of stagnation in Japan after the downturn
in 1991/92 are standard examples of this.
Economic policy in a depression regime: Japan since 1991
These ideas are central to the view that for an economy to get out of a
depression due to slack demand, more than “structural reforms” (reforms
at the microeconomic level) is needed. Japan’s economy has been more
or less stagnating for almost two decades after having experienced record
high economic growth from World War II until 1992. There has been broad
agreement that many structural problems in Japan, not least in the financial
sector, played a role. But would improvements in this regard be able to
drag the economy out of the swamp within a reasonable period of time?
Not according to those macroeconomists who saw deficient effective demand
to be the crucial barrier (Krugman 1998, Svensson 2003). Indeed, the rate
of utilization of productive capacity in Japanese firms had fallen to a very
low level. In 1990 only fourteen percent of Japanese firms responded (in a
repeated survey) that they had excess capacity. In the years 1992-2002 the
number fluctuated around almost fifty per cent (Bank of Japan 2002).
The diagnosis suggested by Krugman and Svensson was that a demand
crisis was the basic problem. This demand crisis shared several characteristics
with the Great Depression in the US in the 1930s − among other things
the presence of deflation (although, after all, the 0.5−2.0 per cent annualdeflation in Japan in several years was considerably less than the 10 per cent
in the US in the period 1930-32). The deflation in Japan went hand in hand
with a short-term nominal interest rate close to its lower bound, zero. This
liquidity trap entailed that conventional monetary policy could not decrease
the short-term nominal interest rate further. A real interest rate significantly
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above the level needed for recovery was the result.
The conclusion by Krugman and Svensson was that without a downward
adjustment of both the short-term and the long-term real interest rate (which
requires creating expected inflation over a time span of ten to fifteen years),
Japan would not get well in a foreseeable future. Others (including Bank
of Japan) had, according to Krugman and Svensson, become hypnotized
by the “structural problems” in Japan and were guilty of a misdiagnosis of
Japan’s stagnation. Structural problems made the productivity level lower
than otherwise, but they could hardly explain the absence of growth. The
structural problems in Japan were barely less pronounced before 1990, yet
the country experienced in the period 1960-90 an average growth rate in
GDP per working hour of five per cent per year. Both India and China
undoubtedly have plenty of structural problems, nevertheless these countries
have since the beginning of the 1980s had per capita growth rates on the
order of magnitude of five to nine per cent per year (source?).5
30.3 Vicious and virtuous circles
As already hinted at, a characteristic feature of the Keynesian approach to
business cycle fluctuations is the emphasis on the sometimes vicious, some-
times virtuous circles that arise, due to production being in the short term
demand-determined rather than supply-determined. A vicious circle may for
example come about in the following way.
Suppose that during an economic boom a housing price bubble evolves.
Sooner or later the bubble bursts, collateral for bank loans loose value (the
balance sheet channel), defaults occur, confidence is shaken, credit is squeezed,
and further defaults occur.6 The financial crisis spills over as an adverse de-
mand disturbance leading to a contraction of production and employment.
The fired workers with less income buy fewer consumption goods (in par-
ticular fewer durable consumption goods). The process tends to be self-
reinforcing in that the fear of being fired increases precautionary saving.
Thus seeing their demand curves continue the inward movement, firms
cut production further. The utilization rate of capital equipment falls and
so does average and marginal . The fall in consumption is thus not offset
by firms’ investment being stimulated, rather the opposite. Firms’ access to
credit is cut down further as the balance sheets deteriorate. An economic
5Paul Krugman’s The Return to Depression Economics (Krugman 2000) reflects on
the need for macroeconomic theory to include depression economics as one of its concerns.6Below we elaborate on the terms in italics.
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30.3. Vicious and virtuous circles 1057
recession or depression may develop if not offset by countercyclical monetary
and/or fiscal policy.
There are several self-reinforcement mechanisms that bring these “circles”
forth, whether they are negative, as above, or positive. Below we list six
examples of such mechanisms. We describe them in their negative mode,
that is, when they lead to vicious circles. They could just as well, however,
be described in their positive mode as when they lead to virtuous circles and
thereby a boom.
Some of the mechanisms have already been touched upon above.
1. The spending multiplier (Kahn 1931, Keynes 1936). A decrease in an
autonomous demand component leads to a decrease in production and
income, and this further reduces demand. The spending multiplier is
larger in a depression, especially in a liquidity trap. Households’ and
firms’ precautionary saving (see Section 30.4) aggravates the down-
turn.7
2. Destabilizing price flexibility (Keynes, Mundell, Tobin). Given there is
some nominal price and wage rigidity, more flexibility may be destabi-
lizing. Suppose there is an adverse shock to investor’s and firms’ general
long-term confidence and that this leads to a downturn of investment
and aggregate demand, production, and employment. Inflation and ex-
pected inflation also go down. In this scenario, is high price flexibility
a good or a bad thing? In fact under a passive (monetarist) monetary
policy (the percent rule) high price flexibility may turn the incipient
recession into a downward wage-price spiral rather than a transitory
dip. This is because opposing effects on aggregate demand are in play.
On the one hand, the fall in inflation increases real money supply and
lowers the nominal rate of interest, thereby stimulating aggregate de-
mand. In an open economy net exports are stimulated. On the other
hand, the fall in expected inflation raises the real rate of interest,
= + −
for a given short-term nominal rate of interest (the policy rate) and a
given interest differential, ≥ 0, thereby reducing demand. Dependingon the circumstances this effect may be the strongest and lead to a self-
sustaining economic contraction. In particular this may happen, when
the nominal rate of interest is already low and therefore near its floor,
7Formally, amultiplier is the ratio of a change in an endogenous variable, here output or
employment, to the change in an exogenous variable, for example autonomous government
spending.
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the zero bound. Then the economy has got into a liquidity trap in
the sense that conventional expansionary monetary policy is no longer
effective (see Svensson 2003).8
Also under a countercyclical monetary policy, the economy may end
up in a liquidity trap. Even though a Taylor rule like (30.7) is gen-
erally considered more stabilizing than a monetarist rule maintaining
a constant growth rate of the money supply, a Taylor rule does not
preclude ending up in a liquidity trap a large adverse demand shock
occurs. Thus the zero lower bound on the nominal interest rate also
implies a limit to the effectiveness of a Taylor rule.
3. The bank lending channel (Bernanke and Blinder, 1988, 1992). If an
economic downturn is on the way, banks may perceive that the riskiness
of loans has increased. A credit squeeze vis-a-vis other banks and the
non-bank public may result and the spread between the interest rate
in the money market and the interest rate that the ultimate borrowers
must pay is increased. This limits capital investment and spending on
durable consumption goods, thus reinforcing the economic downturn.
4. Borrowers’ balance sheet channel (Kiyotaki andMoore, 1997, Bernanke
et al., 1999, Eggertsson and Krugman, 2012). An adverse shock reduces
the net worth of credit-constrained agents (entrepreneurs and house-
holds), whose assets serve as collateral for loans. If expected to persist,
the reduced net worth leads to a credit contraction. In need of liquid-
ity some agents are forced to sell illiquid assets at “fire sale” prices,
thereby further reducing the net worth of debtors. The reduced credit
worthiness leads to less borrowing and less capital investment and con-
sumption next period. Thus aggregate demand falls. The expectation
of this worsening of future market conditions reduces net worth today
further.
5. Coordination failures and multiple equilibria. There are circumstances
(e.g., “spillover complementarity”) where more than one general equi-
librium is possible. Universally held pessimistic expectations lead to
prudent actions that sum to a low-level outcome, thus confirming the
8Nominal interest rates cannot fall below zero, since potential lenders would then prefer
holding cash rather than assets paying a negative interest rate.
The scenario described may take the even more pregnant form of a deflationary spiral
leading to ever-widening financial crisis. The Great Depression in the US in the 1930’s is
a conspicuous example and the problems in Japan since 1991 also have affinity with this.
A simple model of a dynamic liquidity trap and deflationary spirals is presented in Groth
(1993).
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30.4. Precautionary saving 1059
pessimistic expectations. Had all agents held optimistic expectations
they would have made confident upbeat decisions, aggregate demand
would boom and confirm the expectations that brought it about in the
first place (see Heller 1986, Kiyotaki 1988, Xiao, 2004).
6. Hysteresis. The described demand-side dynamics may interact with
the supply side. This occurs when the initial creation of unemploy-
ment, through the de-qualification effect on the unemployed or through
insider-outsider wage-setting behavior, turns a spell of unemployment
into long-term unemployment. Such a phenomenon is called hysteresis.
The technical definition is that hysteresis in unemployment occurs, if
unemployment in the medium term depends positively on unemploy-
ment in the short term.9 This has implications for the trade-off between
short-run benefits of a deficit-financed expansionary fiscal policy in a
liquidity trap and long-run costs in the form of fiscal sustainability
problems arising from a higher government debt.
One factor contributing to the vicious circles under the headings 1 and 5
is the phenomenon of precautionary saving to which we now turn.
30.4 Precautionary saving
In the first years after the crash at the New York stock exchange in 1929 a
sharp fall in private consumption and investment occurred. Many economists
argue that this should be seen in the light of the fact that the consump-
tion/saving decision is sensitive to increased uncertainty.10 Similarly, the in-
ternational financial crisis, triggered by the subprime mortgage crisis in the
US in 2007, created a massive worldwide economic recession 2008-2010 (the
“Great Recession”). In this downturn precautionary saving is again likely to
have played an important role. If people suddenly feel more uncertain about
what is going to happen, they tend to be more prudent and increase their
saving in order to have a “buffer-stock”. But this may aggravate the negative
spiral of falling aggregate demand and production.
To clarify the issue, we first consider a simple model of a household’s
consumption/saving decision under uncertainty. Second, we discuss the pos-
sible macroeconomic implications and relate the discussion to the different
business cycle “schools”. Indeed, whether one includes precautionary saving
9See Blanchard (1990). A corresponding virtuous hysteresis can arise through the
qualification or learning-by-doing effect of being employed. More generally on hysteresis,
see Fiorillo (1999).10Romer (1990) provides an analysis.
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among the factors that can reinforce a business-cycle downturn depends very
much on the basic conception of business cycles (New Classical or Keynesian,
supply-side economics or demand-side economics).
30.4.1 Consumption/saving under uncertainty
Consider a given household facing uncertainty about future labor income and
capital income. For simplicity, assume the household supplies one unit of
labor inelastically each period. The household never knows for sure whether
it will be able to sell that amount of labor in the next period. Given the time
horizon ≥ 2, the decision problem is:
max0(0) = 0[
−1X=0
()(1 + )−] s.t. (30.8)
≥ 0 (30.9)
+1 = (1 + ) + − 0 given, (30.10)
≥ 0 (30.11)
where 0 0 and 00 0 (so there is risk aversion). The rate of time prefer-ence w.r.t. utility is −1 (usually 0 seems realistic, but here the signof is not important). We think of “period ” as the time interval [ + 1)
Hence, the last period within the planning horizon is period − 1 Realfinancial wealth is denoted and ( 0) is the real wage, whereas is the
exogenous amount of employment offered to the household by the labor mar-
ket in period , 0 ≤ ≤ 111 The (net) real rate of return on financial wealthis called ( −1) The symbol 0 stands for the expectation operator, con-ditional on the information available in period 0. This information includes
knowledge of all variables up to period 0, including that period. There is
uncertainty about future values of , and , but the household knows
the stochastic processes that these variables follow.12 The risk associated
with the uncertainty is realistically assumed to be uninsurable.
There are two endogenous variables, the control variable and the state
variable The constraint (30.9) defines the “control region”, whereas (30.10)
is the dynamic budget identity, and (30.11) is the solvency condition, given
the finite planning horizon . The decision as seen from period 0 is to choose
a concrete action 0 and a set of contingent plans ( ) about what to do
11More generally, could be replaced by interpreted as any kind of exogenous
income, say an uncertain pension.12Or at least the household has beliefs about these processes and calculates subjective
conditional probability distributions on this basis.
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30.4. Precautionary saving 1061
in the future periods, = 1 2 − 1 This decision is made so that ex-pected discounted utility, 0(0) is maximized. We call the function ( )
a contingent plan because it tells what consumption will be in period , de-
pending on the realization of the as yet unknown variables up to period
in particular the state variable To choose the action 0 in a rational way,
the household must take into account the whole future, including what the
optimal conditional actions in the future will be.
In every new period the household solves the remainder of the problem
as seen from that period. Defining ≡ (1 + ), the remainder of the
problem as seen from period ( = 0 1 ) is: max =
() + (1 + )−1[(+1) + (+2)(1 + )−1 + ] (30.12)
s.t. (30.9)-(30.11), given.
To solve the problem we will use the substitution method. First, from (30.10)
we have
= (1 + ) + − +1 and (30.13)
+1 = (1 + +1)+1 + +1+1 − +2
Substituting this into (30.12), the problem is reduced to an essentially uncon-
strained maximization problem, namely one of maximizing the function
w.r.t. +1 +2 (thereby indirectly choosing +1 −1) Hence,we first take the partial derivative w.r.t. +1 in (30.12) and set it equal to
0:
+1= 0() · (−1) + (1 + )−1[
0(+1)(1 + +1)] = 0
Reordering gives the stochastic Euler equation,
0() = (1+)−1[
0(+1)(1++1)] = 0 1 2 −2 (30.14)These first-order conditions describe the trade-off between consumption
in period and period + 1, as seen from period The optimal plan must
satisfy that the current utility loss by decreasing consumption by one unit
is equal to the discounted expected utility gain next period by having 1++1extra units available for consumption, namely the gross return on saving one
more unit. This holds for = 0 1 2 −2 In the final period, the decisionmust be to consume everything left:
−1 = (1 + −1)−1 + −1−1 (30.15)
since it is not optimal to end up with 0 (indeed, the transversality
condition is that = 0)
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2c
2c
bc *bcac*
ac
bc*ac
'u
)( 2cu
u
)(' 2cu
)(' 2*1 cuE
)(' 21 cuE
c
c
Figure 30.1
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
30.4. Precautionary saving 1063
But first-order conditions only tell us about the relative levels of con-
sumption over time. The absolute level of consumption is determined by the
condition that the initial level of consumption must be the highest possible
0 consistent with, first, (30.14) for = 0 second, (30.14) in expected value,
as seen from period 0, for = 1 2 − 2; and, third, (30.15) in expectedvalue as seen from period 0.
Let us first consider the simplest case.
Risk-free rate of return
In this case, +1 is known and there is only uncertainty about future labor
income. Hence, (30.14) reduces to
0 () =1 + +1
1 + [
0 (+1)] = 0 1 2 − 2 (30.16)
It is natural to assume that higher wealth is associated with lower (or at
least not higher) absolute risk aversion (i.e., not higher values of −000). Inthat case, it can be shown that marginal utility 0 is a strictly convex functionof that is, (0)00 0. But this implies that increased uncertainty in the
form of a mean-preserving spread will lead to lower consumption “today”
(more saving) than would otherwise be the case. This is what precautionary
saving is about.
Fig. 30.1 gives an illustration. We can choose any utility function with
(0)00 0 The often used logarithmic utility function is an example since
() = ln gives 0() = −1, 00() = −−2 and 000() = 2−3 0. In the
figure it is understood that = 3 and that we consider the decision problem
as seen from period 1 There is uncertainty about labor income in period
2. It can be because the real wage is unknown or because employment is
unknown or both. Suppose, for simplicity, that there are only two possible
outcomes for labor income (≡ ) say and each with probability12. That is, given 2, there are, in view of (30.15), two possible outcomes for
2:
2 =
½ = (1 + 2)2 + with probability = 1
2
= (1 + 2)2 + with probability = 12.
(30.17)
Mean consumption will be = (1 + 2)2 + where = 12( + ).
Suppose 1 is chosen optimally. Then, with = 1 (30.16) is satisfied,
and 2 is given, by (30.10) with = 1. The lower panel of Fig. 30.1 shows
graphically, how 10(2) is determined, given this 2. In case of higher
uncertainty in the form of a mean-preserving spread, i.e., a higher spread,
| − |, but the same mean the two possible outcomes for 2 are ∗ and
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)(' cu
Ec
)(' Ecu
))('( cuE
1c
Case 1
)(' cu
Ec
)('))('( EcucuE
Case 2
c
c
Figure 30.2
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.
30.4. Precautionary saving 1065
∗ , if 2 is unchanged and, hence, unchanged. Then, the expected marginalutility of consumption becomes greater than before, as indicated by 1
0(∗2)in the figure. In order that (30.16) can still be satisfied, a lower value than
before of 1 must be chosen (since 00 0), hence, more saving.
True enough, this increases 2 so that the expected value of 2 is in fact
larger than on the figure. Hereby the new 10(2) ends up somewhere
between the old 10(2) and 1
0(∗2) in the figure. The conclusion is stillthat the new 1 has to be lower than the original 1 in order that the first-
order condition (30.16) can be satisfied in the new situation.
This phenomenon is called precautionary saving. To be more precise, we
define precautionary saving as the increase in saving resulting from increased
uncertainty. In the above example, increased uncertainty (a mean-preserving
spread) implied lower consumption “today”, that is, precautionary saving.
Consumption is postponed in order to have a buffer-stock. The intuition is
that the household wants to be prepared for meeting bad luck, because it
wants to avoid the risk of having to end up starving (“save for the rainy
day”).
Note that the mathematical background for the phenomenon is the strict
convexity of marginal utility, i.e., the assumption that (0)00 0 This implies(0()) 0() in view of Jensen’s inequality (see Appendix). Case 1 inFig. 30.2 shows the example () = ln i.e., 0() = −1If instead, (0)00 = 0 as with a quadratic utility function, then the graph
for 0(2) is a straight line (cf. case 2 in Fig. 30.2), and then precautionarysaving can not occur. Indeed, a quadratic utility function can be written
() = − 122, 0 “large”. (30.18)
Then 0() = − , a straight line. By “large” is meant “large relative to
the likely levels of consumption” so that only the upward-sloping branch of
the function becomes relevant in practice (thus avoiding a negative 0()).This would be an example of so-called certainty equivalence. We say that
certainty equivalence is present, if the decision under uncertainty follows the
same rule as under certainty, only with actual values of the determining
variables replaced by the expected values. The easiest case is to compare
a situation where the relevant exogenous variables take on their expected
values with a probability one (certainty) and a situation where they do that
with a probability less than one (uncertainty). If the decision is the same
in the two situations, certainty equivalence is present. So, when there is
certainty equivalence, the decision under uncertainty is independent of the
size of the uncertainty, measured by, say, the variance of the relevant exoge-
nous variable(s). Quadratic utility implies certainty equivalence. Yet, since
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(30.18) gives 00 = −1 0 a household with quadratic utility is risk averse.Hence, for precautionary saving to arise, more than risk aversion is needed.
What is needed for precautionary saving to occur is 000 0 i.e., “pru-
dence”. Just as the degree of (absolute) risk aversion is measured by −000(i.e., the degree of concavity of the utility function), the degree of (absolute)
prudence is measured by −00000 (i.e., the degree of convexity of marginalutility). The degree of risk aversion is important for the size of the required
compensation for uncertainty, whereas the degree of prudence is important
for how the household’s saving behavior is affected by uncertainty.
Uncertain rate of return
We have just argued that strictly convex marginal utility is a necessary con-
dition for precautionary saving. But it is not a sufficient condition. This is
because there may be uncertainty not only about future labor income, but
also about the rate of return on saving.
Consider the case where, as seen from period , +1 is unknown. Then
the relevant first-order condition is (30.14), not (30.16). Now, at least at
the theoretical level, the tendency for precautionary saving to arise may
be dampened or even turned into its opposite by an offsetting factor. For
simplicity, assume first that there is no uncertainty associated with future
labor income so that the only uncertainty is about the rate of return, +1
In this case it can be shown that there is positive precautionary saving if the
relative risk aversion, −000, is larger than 1 (“it is good to have a bufferin case of bad luck”) and negative precautionary saving if the relative risk
aversion is less than 1 (“get while the getting is good”).
It is generally believed that the empirically relevant assumption from a
macroeconomic point of view is that −000 1 Thus, increased uncer-
tainty about the rate of return should lead to more saving. The resulting
precautionary saving then adds to that arising from increased uncertainty
about future labor income.
30.4.2 Precautionary saving in a macroeconomic per-
spective
Simple calculations as well as empirical investigations (for references, see
Romer 2001, p. 357) indicate that precautionary saving is not only a theo-
retical possibility, but can be quantitatively important. A sudden increase in
perceived uncertainty seems capable of creating a sizeable fall in consump-
tion expenditure (in particular expenditure on durable consumption goods)
and thereby in aggregate demand. According to a study by Christina Romer
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30.5. Literature notes 1067
(1990) this played a major role for the economic downturn in the US after
the crash at the stock market in 1929 (see also Blanchard, 2003, p. 471 ff.).
Note that the conception of precautionary saving as an important busi-
ness cycle force does not fit equally well in all business cycle theories. In
new-classical theories (since the 1980s the RBC theory) a lower propensity
to consume is immediately and automatically compensated by higher invest-
ment demand (in a closed economy). According to the RBC model from
the previous chapter, aggregate demand continues to be sufficient to absorb
output at essentially full capacity utilization. Higher uncertainty just leads
to a change in the composition of demand, a manifestation of Say’s law.
Keynesians consider this story to be contradicted by the data. Less con-
sumption spending seems far form being automatically offset by higher in-
vestment spending. Instead, vicious and virtuous circles are emphasized,
these phenomena arising from production being in the short term demand-
determined rather than supply-determined. An adverse shock, say a bursting
housing market bubble, will, through precautionary saving, lead to a contrac-
tion of demand and therefore a downturn of production.
Also firms’ behavior may in an economic crisis have aspects of precau-
tionary financial saving. A deep crisis generates a lot of uncertainty: firms
do not understand what has happened and no one knows what actions to
choose. The natural thing to do is to pause and wait until the situation
becomes clearer. This entails a cutback in the plans for further purchase
of investment goods. So on top of households’ precautionary saving we have
prudent investment behavior by the firms.
30.5 Literature notes
(incomplete)
The self-fulfilling prophesy investment theory by Kiyotaki (1988) and the
inventory investment theory by Blinder ( ) are examples of business cycle
theory emphasizing firms’ investment.
30.6 Appendix
Jensen’s inequality
Jensen’s inequality is the proposition that when is a stochastic variable,
and the function is convex, then
() ≥ ()
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with strict inequality, if is strictly convex (unless with probability 1 is
equal to a constant). It follows that if is concave (i.e., − is convex), then
() ≤ ()
with strict inequality, if is strictly concave (unless with probability 1 is
equal to a constant).
30.7 Exercises
c° Groth, Lecture notes in macroeconomics, (mimeo) 2013.