Chapter 28 Business cycle ﬂuctuations - ku and lecture... · Chapter 28 Business cycle ﬂuctuations This chapter presents stylized facts and basic concepts relating to business

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).

975

976 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”.

c°C. Groth, Lecture notes in macroeconomics, (mimeo) 2012

28.2. Key terms from the business cycle vocabulary 977

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.

Table 1 Glossary concerning shocks and their effects



Shock type

Effect on dependent variable Temporary Fading Persistent

Temporary

Fading

Persistent

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

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

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.


28.3. A quick glance at the Great Recession and its aftermath 979

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

laureate lecture, Dale Mortensen (2011) concluded: “The real problem is

that demand for goods and services has not recovered because real interest



Figure 28.2:


28.3. A quick glance at the Great Recession and its aftermath 981

Figure 28.3:



Figure 28.4:


28.4. Conclusion 983

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




Chapter 29

The real business cycle theory

Since the middle of the 1970s quite different approaches to the explanation

of business cycle fluctuations have been pursued. We may broadly classify

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

attempts 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

to 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.

985

986 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. Government intervention

with the purpose of stabilization is seen as likely to be counterproductive.

Given the uncertainty due to shocks, the market forces establish a Pareto op-

timal moving equilibrium. “Economic fluctuations are optimal responses to

uncertainty in the rate of technological change”, as Edward Prescott claims

(Prescott 1986).

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 much broader

class of quantitative models, including models with an emphasis on money,

nominal price stickiness, and demand shocks. The “economic philosophy”

behind the RBC theory in the sense the term is used here, was proclaimed

in Prescott (1986).

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.


29.1. A simple RBC model 987

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)



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 2.3 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)4 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 Appendix). The difficulty with this

specification is that it tends to generate too little fluctuation in employment

3We recall that a sequence of stochastic variables with zero mean, constant variance,

and zero covariance across time is called white noise.4This 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.



and output. This is because, when shocks are permanent, large wealth effects

offset the intertemporal substitution in labor supply.

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

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.5 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.

5Concavity implies adding the assumption 1122 − (12)2 ≥ 0



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 + ),6 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

6Multiplying a utility function by a positive constant does not change the associated

optimal behavior.



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 condition7

+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−+)+]

7Generally speaking, for a given differentiable function (1 ) where is a

stochastic variable and 1 are parameters, we have

(( ))

=

( )

= 1



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.8 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

8In 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.



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)


29.2. A deterministic steady state 995

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)



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 10.9 Existence of general equilibrium in our Ramsey framework requires

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.10 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)

9King and Rebelo, 1999, p. 947, express this in terms of the growth-adjusted discount

factor ≡ (1 + )−11− so that 1 + ∗ = (1 + ) = 10Since 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.


29.3. On the approximate solution and numerical simulation 997

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).11 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.

11Alternatively, 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.



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



− = − ≡ ()

−

= −

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 12 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

12This is not surprising, since reflects the desire for leisure smoothing across time.



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



such an approximate dynamic system analytically.13 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 residual14 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.15 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

13For details one may consult Campbell (1994, p. 468 ff.), Obstfeld and Rogoff (1996,

p. 503 ff.), or Uhlig (1999).14Given (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

15King and Rebelo (1999, p. 982-993) believe that the problem can be overcome by

refinement of the RBC model.



the shocks can be the driving force behind business cycles.16

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

16At 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).


29.4. The two basic propagation mechanisms 1003

= 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



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


29.6. Conclusion 1005

intertemporal elasticity of substitution in labor supply17 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).18 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’ quits19

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 Conclusion

It seems 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

17Recall 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 5.18Handbook of Labor Economics, vol. 1, 1986, Table 1.22, last column. See also Hall

(1999, p. 1148 ff.).19See Chapter 28.



other hand the quantitative methods introduced by Lucas, Prescott, and oth-

ers. A significant amount of recent research on business cycle fluctuations has

left the RBC theory, but apply similar quantitative methods. These methods

are now often summed up under the heading DSGE models. This approach

consists in an attempt at building small quantitative Dynamic Stochastic

General 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).

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).


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).


29.8. Appendix: Technological change as a random walk with drift 1007

29.8 Appendix: Technological change as a ran-

dom 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-

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 020 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.

29.9 Exercises

20The 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




Chapter 30

Keynesian perspectives on

business cycles

Results from a thorough time series analysis of the US economy are summa-

rized in the following way by Blanchard (1989):

(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 is an example of a Keynesian-style interpretation of macroeconomic

fluctuations. Point (a) indicates that demand shocks, such as a shift in the

state of confidence, a shift in government spending, or a shift in liquidity

preference 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 major 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 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

arise when production is demand-determined rather than supply-determined.

One may say that the core of Keynesian thinking is the refutation of Say’s

law which says that “aggregate supply creates its own aggregate demand”

1009

1010

CHAPTER 30. KEYNESIAN PERSPECTIVES ON BUSINESS

CYCLES

or that “income is automatically spent on produced goods and services”.

As described in Chapter 19, Keynes’ refutation of this doctrine rests on a

rejection of the validity of the Walrasian budget constraint and Walrasian

demands and supplies when trade occurs outside Walrasian equilibrium and

a replacement of these concepts by the notions 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 recent financial and economic crisis

we have witnessed a fresh outbreak of disagreements between Keynesians

and new-Classicals. This comes into view both at the theoretical level and

in regard to opinions about economic policy.

In this book we use the label “Keynesian” when relevant to bring to

light fundamental differences vis-a-vis classical (or Walrasian) macroeco-

nomic thinking 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 which generally makes effective demand the

binding constraint. Given the preset , firm ’s ex post production level

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

1A more specific new-Keynesian “workhorse model” is described in Chapter 32.


30.1. The short run 1011

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

many 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

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,


1012


CYCLES

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. 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 short-term 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

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 per-

spective changes in expectations are important. But expectation formation


30.2. From the short to the medium run 1013

is a difficult field and far from settled. The Keynesian approach is not inter-

locked with any specific hypothesis about expectation formation. Depending

on the circumstances, alternative hypotheses are introduced, including ra-

tional expectations, adaptive expectations, extrapolative expectations, and

adaptive learning.

There is among theorists an increasing awareness/recognition that mod-

els 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, according to the actual economic evolution.

Example: for a decade up to 2006/07 house prices in many countries were

rising much faster than the economy and faster than the construction costs.

House prices seemed systematically diverging from the fundamental value. In

such a situation some people begin to recognize that the evolving explosive

price path in the housing market is not sustainable. But at the same time

many market participants feel a deep-seated uncertainty about when the tip-

ping 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 right 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

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),

2See, e.g., Blanchard and Katz (1999).


1014


CYCLES

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 as implied by (??). Then roughly

the price inflation rate equals the wage inflation rate minus the productivity

growth rate,

− −1 = − −1 −

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.

3See Hendry (2008).



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

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

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.


1016


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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 central bank’s implicit real

interest rate offered to the ultimate borrowers when = ∗ and =

including the interest differential ≥ 0 reflecting risk. The Taylor rule

thus gives the target nominal interest rate chosen by the central bank as

an increasing linear function of the “output gap”, − ∗ and the excessinflation 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. Return-

ing 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 MC for any given output level

becomes 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

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 per cent 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 thepresence of deflation (although, after all, the 0.5−2 per cent annual deflationin 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 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


1018


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per working hour of five per cent per year (Jones 1998). 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

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

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.


30.3. Vicious and virtuous circles 1019

and income, and this further reduces demand. Households’ and firms’

precautionary saving (see Section 30.4) aggravates the downturn.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,

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 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

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.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).


1020


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liquidity trap if there is a large adverse demand shock. 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, 2010). 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

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

9See Blanchard (1990). A corresponding virtuous hysteresis can arise through the


30.4. Precautionary saving 1021

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

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

qualification or learning-by-doing effect of being employed. More generally on hysteresis,

see Fiorillo (1999).10Romer (1990) provides an analysis.


1022


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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

market in period , 0 ≤ ≤ 111 The (net) real rate of return on financialwealth is called ( −1) The symbol 0 stands for the expectation oper-ator, conditional 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

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

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

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.



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)

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.


1024


CYCLES

2c

2c

bc *bcac*

ac

bc*ac

'u

)( 2cu

u

)(' 2cu

)(' 2*1 cuE

)(' 21 cuE

c

c

Figure 30.1:



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

∗ , 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.


1026


CYCLES

)(' cu

Ec

)(' Ecu

))('( cuE

1c

Case 1

)(' cu

Ec

)('))('( EcucuE

Case 2

c

c

Figure 30.2:



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

(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.


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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 from this source.

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

(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 business

cycle force does not fit equally well in all business cycle theories. In new-

classical theories (nowadays the RBC theory), a lower propensity to consume

is immediately and automatically compensated by higher investment demand

(in a closed economy). According to classical and new-classical theory, ag-

gregate demand continues to be sufficient to absorb output at full capacity

utilization (full employment of both the capital stock and the labor force).

Higher uncertainty only leads to a change in the composition of demand.


30.5. Literature notes 1029

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 cut back the plans for further purchase of

investment goods. So on top of households’ precautionary saving we have

prudent investment behavior by the firms.


(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

() ≥ ()

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


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Chapter 28 Business cycle ﬂuctuations - ku and lecture... · Chapter 28 Business cycle ﬂuctuations This chapter presents stylized facts and basic concepts relating to business