THEORIES OF FLUCTUATIONS

Lecture notes prepared for the course:Advanced Macroeconomics (mod.2)

Master of Science in EconomicsUniversity of Pisa - Scuola Superiore Sant’Anna

Alessio MONETA1

April 29, 2015

Contents

1 Introduction to macroeconomic fluctuations 3

1.1 Some key facts about fluctuations . . . . . . . . . . . . . . . . . . . . . 3

1.2 Measuring economic fluctuations . . . . . . . . . . . . . . . . . . . . . . 10

1.3 What causes economic fluctuations? . . . . . . . . . . . . . . . . . . . . 11

2 Real-Business-Cycle Theory 12

2.1 Tenets of the RBC school . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Basic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 The basic RBC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Household behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Solving the model in a special case . . . . . . . . . . . . . . . . . . . . . 21

3 Keynesian Theories of Fluctuations 23

3.1 The IS/LM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 The goods market: the IS curve . . . . . . . . . . . . . . . . . . 23

3.1.2 The money market: the LM curve . . . . . . . . . . . . . . . . . 25

3.1.3 The relationship between real and nominal interest rates . . . . . 28

3.1.4 The interplay between the IS and LM curve . . . . . . . . . . . . 28

1Institute of Economics, Scuola Superiore Sant’Anna, Piazza Martiri della Liberta 33, 56127 Pisa, Italy.Email: amoneta@sssup.it. The author is very grateful to be informed of any mistake found in this text!

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1 Introduction to macroeconomic fluctuations

In this section we address three questions: (i) what are macroeconomic fluctuations? (ii)how do we measure them? (iii) what are their causes?

1.1 Some key facts about fluctuations

Understanding why the economy fluctuates over time is perhaps one of the main tasksof macroeconomics. According to Snowdon and Vane (2005, p. 304), for example, thecentral goal of macroeconomics is to provide “coherent and robust explanations of aggre-gate movements of output, employment and the price level, in both the short run and thelong run”. But before examining the causes of these phenomena, we have to identify theirtypical features and find out how they can be quantitatively described.

When we talk about economic fluctuations or business cycles2 the focus is usuallyon the short run (few months) or middle run (few years).The focus on the short run wasparticularly emphasized by J.M. Keynes: “In the long run we are all dead. Economistsset themselves too easy, too useless a task if in tempestuous seasons they can only tell usthat when the storm is long past the ocean is flat again” (Keynes, 1923). When Keyneswrote these lines, his polemical target was the quantity theory of money and the connectedprinciple that money is neutral3 in the long run. Classical economists (e.g. Ricardo) wereconcerned with long-run equilibrium. Keynes’s analysis, which culminated with the pub-lication of the General Theory (Keynes, 1936), shifted the emphasis to the short-run, andto out-of-equilibrium states of the economy. In his work Keynes aimed to explain thecauses of economic instability since he argued that capitalist market economies can stayat less than full employment for prolonged period of time, and the Great Depression wit-nessed that this was indeed possible. However, it would be incorrect to say that Keyneswas just interested in the short run. The importance of long-term consequences of eco-nomic decisions was duly recognized by Keynes (cfr. Keynes, 1930). But it is true that itsemphasis on instability and out-of-equilibrium phenomena reconsidered the importanceof a careful analysis of the short-run fluctuations.

While Keynes advocated to enrich economic theory with the analysis of economicfluctuations, at the same time other, less theoretical, approaches emerged. In the U.S. theNational Bureau of Economic Research (a private nonprofit research organization) wasfounded in 1920. The first director of research was Wesley Mitchell who pioneered em-pirical research of business cycle phenomena. In 1946 Arthur Burns (chair of the FederalReserve form 1970 to 1978) and Wesley Mitchell (an important figure in the Americaninstitutionalism) published a book (Measuring Business Cycles), which contains the fol-lowing definition of business cycles:

2Economic fluctuations or business cycles? Which is the right term and what is the difference amongthem? They are synonymous. Economists use these two terms interchangeably. Those who want to em-phasize the regular feature of the ups and downs of the economy use the term “business cycle.” The term“economic fluctuations” is perhaps more neutral.

3Money is neutral if after a monetary expansion (increase in the supply of money) the price level, nomi-nal wages and the nominal interest rate will increase but all the real values (output produced, consumption,investment, etc.) remain the same.

3

“Business cycles are a type of fluctuations found in the aggregate eco-nomic activity of nations that organise their work mainly in business enter-prises. A cycle consists of expansions occurring at about the same time inmany economic activities, followed by similarly general recessions, contrac-tions, and revivals which merge into the expansion phase of the next cycle:this sequence of changes is recurrent but not periodic, in duration businesscycles vary from more than one year to ten or twelve years.” (Burns andMitchell, 1946)

The main features of a business cycle are therefore: (i) alternation in the state (upsand downs) of the economy; (ii) a recurrence of the ups and downs, although not per-fectly periodic; (iii) a rough coherence between different measures of the economy.

Figure 1: Romer D. (4ed 2012) Advanced Macroeconomics, Mc Graw Hill, Figure 5.1.

Figure 1 (from Romer, 2012) plots seasonally adjusted real GDP quarterly from 1947to 2009, in log scale. We see that there is an almost linear (in log scale!) dominant path,which we call the trend, and fluctuation around it, i.e. the business cycle. We also see thatthe recurrence of ups and downs does not display a regular cycle, but the alternation doesnot appear to be completely random either. Before looking at some stylized facts abouteconomic fluctuations, let us introduce some useful terminology (cfr. Hoover, 2012, pp.143-144).

Trend: dominant path of a macroeconomic time series indicating economic activity(e.g. real GDP).

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Cycle: fluctuations around this trend alternating peaks and troughs

Recession (synonyms: slump, contraction, bust): period between peak and trough.

In the USA, the NBER is generally seen as the authority for dating US recessions.The Business Cycle Dating Committee of the NBER provides a formal dating of US re-cessions. The dates are usually announced with a certain lag. Its definition of recessionis “significant decline in economic activity spread across the economy, lasting more thana few months, normally visible in real GDP, real income, employment, industrial pro-duction, and wholesale-retail sales.” Notice that the NBER’s decision is not the result ofany formal algorithm but is ultimately based on overall impressions on the movements ofmany economic indicators.4

In Europe the Centre for Economic Policy Research, a think tank based in UK, definesEuropean recession as “a significant decline in the level of economic activity, spreadacross the economy of the euro area, usually visible in two or more consecutive quarters ofnegative growth in GDP, employment and other measures of aggregate economic activityfor the euro area as a whole.” 5

Expansion (synonyms boom, recovery): period between trough and peak.

Depression: A particularly sever recession (in terms of percentage change and dura-tion), e.g. Great Depression of 1929-19336; 2007-2009 US Great Recession7; Eurozonecrisis 2009-?

Growth recession (slowdown): period of slower than trend growth

Cycle: (i) period between trough and next trough, or (ii) period between peak andnext peak.

Recovery: (i) period from trough to the level of the previous peak, or (ii) period fromtrough to the level of the trend.

Figure 2 from Hoover (2012) depicts a stylised economic time series

As Romer (2012, Ch. 5.1) points out, we can identify some major facts about eco-nomic fluctuations.

4Cfr. http://www.nber.org/cycles.html5Cfr. http://www.cepr.org/data/dating6In the USA the trough of the Great Depression was technically reached in March 1933, but recovery

was not secure until the beginning of World War II.7Technically the US economy entered a recession in December 2007 and the through was reached in

June 2009. But unemployment continued to rise after that date, reaching 10% in November 2009. BetweenDecember 2007 and June 2009 GDP fell by 4.3 percent, the largest fall during a recession since the GreatDepression 1929-1933.

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Figure 2: A stylised economic time series, from Hoover (2012) Applied IntermediateMacroeconomics, Cambridge University Press, Figure 5.6.

1. No clear and simple regularity in the cyclical pattern

A first important empirical fact about economic fluctuations is that they do not exhibitany strictly regular cyclical pattern. Figure 1 shows that output declines are of differentsize and nature. Table 1 shows that recessions vary in terms of length and changes inouput and unemployment.

Table 1: Recessions in the United States since World War II

Number of months Change in real GDP HighestRecession dates from peak to through from peak to through unemployment rate*

Nov. 1948-Oct. 1949 11 -1.7% 7.9%July 1953-May 1954 10 -2.7 5.9Aug. 1957-Apr. 1958 8 -1.2 7.4Apr. 1960-Feb. 1961 10 -1.6 6.9Dec. 1969-Nov. 1970 11 -0.6 5.9Nov. 1973-Mar. 1975 16 -3.1 8.6Jan. 1980-July 1980 6 -2.2 7.8July 1981-Nov. 1982 16 -2.9 10.8July 1990-Mar. 1991 8 -1.3 6.8Mar. 2001-Nov. 2001 8 -0.5 6.0Dec. 2007 - June 2009 18 -5.1 10

*included the aftermath period of the recessionSource: NBER

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2. Symmetries and asymmetries in output movements

In the U.S. (post-World War II) output growth is distributed roughly symmetricallyaround its mean, that is falls in output are of similar size of rises in output. In the U.S.(post-World War II) the expansion phase is longer (about five times) than the recessionphase. Figure 3 shows U.S. real GDP growth rates (quarterly compound annual and an-nual)8. Notice that mean is positive (around 3.5%).

years

perc

ent p

er y

ear

1950 1960 1970 1980 1990 2000 2010

−10

−5

05

1015

quarterly growth rate(compound annual)

annual growth rate

mean a. growth rate

Real GDP Growth Rates, US 1948−2012

Figure 3: U.S. real GDP Growth rates (quarterly and annual)

8A growth rate is the proportionate (or percentage) rate of change per unit time and is calculated as

Xt =Xt −Xt−1

Xt−1=

Xt

Xt−1− 1.

If data are observed quarterly, then Xt is a quarterly growth rate. For the sake of comparisons it is useful toconvert a quarterly growth rate in annual units. We get the compound annual quarterly growth rate:

Xt =

(Xt

Xt−1

)4

− 1.

The annual growth rate computed from quarterly data is:

Xt =Xt

Xt−4− 1.

We see from Figure 3 that the annual growth rate is smoother than the compound annual quarterly growthrate (see Hoover, 2012, pp. 764-766).

In continuous time the growth rate of X(t) corresponds to the derivative of logX(t). Indeed

X(t) =X(t)

X(t)=dX(t)

dt

1

X(t)=d log(X(t))

dt.

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Table 2: Behaviour of the components of output in recessions (U.S. 1947 - 2009)

Average share in fallAverage share in GDP in recessions

Components of GDP in real GDP relative to normal growth

ConsumptionDurables 8.9% 14.6%Nondurables 20.6 9.7Services 35.2 10.9

InvestmentResidential 4.7 10.5Fixed nonresidential 10.7 21.0Change in inventories 0.6 44.8

Net exports -1 -12.3

Government purchases 20.2 1.3

Source: Romer D. (4ed 2012) Advanced Macroeconomics, Mc Graw Hill, Table 5.2.

3. Fluctuations are distributed very unevenly over the components of output

Table 2 shows the average shares of each of the components of GDP (column 2) and theaverage shares of the same components in the declines of output (relative to the normalgrowth) during recessions. Thus, for example, inventory investment account for the 0.6%of GDP, but its fluctuations account for 44.8% of the shortfall in GDP growth relativeto GDP normal growth. In sum, some components of GDP fluctuates much more thanothers.

4. Co-movements

Another important feature of the business cycles is the co-movements of many eco-nomic variables in a predictable way. This has led Lucas (1977) to claim that “withrespect to the qualitative behaviour of co-movements among series business cycles are allalike”. It suggests “the possibility of a unified explanation of business cycles grounded inthe general laws governing market economies, rather than in political or institutional char-acteristics specific to particular countries or periods” (Lucas, 1977, p.10). It is not obviousto identify general laws in economics, but it is true that there are robust empirical regular-ities about the deviations from trend of many economic variables. Table 3 summarizes thetypical cyclical behaviour of a wide range of key economic variables. Variables that movein the same direction (display postive correlation) as GDP are procyclical; variables thatmove in the opposite direction (display negative correlation) to GDP are countercyclical;variables that display no clear pattern (zero correlation) are acyclical. With respect totiming, variable that move ahead of GDP are leading variables; variables that follow GDPare lagging variables; and variables that move at the same time as GDP are coincidentvariables (Snowdon and Vane, 2005, p. 307).

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Table 3: Co-movements in business cycle

Variable Direction TimingProduction

Industrial production Procyclical Coincident

ExpenditureConsumption Procyclical CoincidentBusiness fixed investment Procyclical CoincidentResidential investment Procyclical LeadingInventory investment Procyclical LeadingGovernment purchase Procyclical -

Labour market variablesEmployment Procyclical CoincidentUnemployment Countercyclical No clear patternAverage labour productivity Procyclical LeadingReal wage Procyclical -

Money supply and inflationMoney supply Procyclical LeadingInflation Procyclical Lagging

Financial variablesStock prices Procyclical LeadingNominal interest rates Procyclical LaggingReal interest rates Acyclical -

Source: Abel, A.B. and Bernanke, B.S. (2001) Macroeconomics, Addison-Wesley, p. 288 and Snowdon B. and Vane, H.R. (2005) Modern Macroeco-nomics, Efward Elgar, p. 306

5. Some regularities in recessions

Another issue is whether output fluctuations have changed their characteristics overtime. About US aggregate fluctuations Romer (2012) expresses the following considera-tions:

“One can think of the macroeconomic history of the United States sincethe late 1800s as consisting of four broad periods: the period before the GreatDepression; the Depression and World War II; the period from the end ofWorld War II to about the mid-1980s; and the mid-1980s to the present. Al-though our data for the first period are highly imperfect, it appears that fluctu-ations before the Depression were only moderately larger than in the periodfrom World War II to the mid-1980s. Output movements in the era beforethe Depression appear slightly larger, and slightly less persistent, than in theperiod following World War II; but there was no sharp change in the characterof fluctuations. Since such features of the economy as the sectoral compo-sition of output and role of government were very different in the two eras,this suggests either that the character of fluctuations is determined by forcesthat changed much less over time, or that there was a set of changes to theeconomy that had roughly offsetting effects on overall fluctuations” (Romer,2012, p. 192)

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Moreover, there are strong regularities in the behaviour of some important macroeco-nomic variables during recessions. During recessions, Employment falls, unemploymentrises and productivity (as measured as output per worker-hour) declines. The Okun’s lawdescribes the empirical correlation between shortfalls in GDP and rises in unemploymentrate. In the original formulation a shortfall in GDP of 3% relative to normal growth isassociated with a 1 percentage-point rise in the unemployment rate; more recently theassociation has been updated to 2:1 (cfr. Romer, 2012, p. 193).

1.2 Measuring economic fluctuations

NBER methodology

The NBER identifies the business cycle turning points in U.S. retrospectively and on anongoing basis. NBER researchers determine these dates using a two-step procedure: (i)local maxima and minima (peaks and troughs) are determined for individual series (withthe help of a computer program, but the ultimate decision is judgemental); (ii) commonturning points across series are identified.

“If, in the judgment of the analysts, the cyclical movements associatedwith these turning points are sufficiently persistent and widespread acrosssectors, than an aggregate business cycle is identified and its peaks and troughsare dated. Currently, the NBER Business Cycle Dating Committee uses dataon output, income, employment, and trade, both at the sectoral and aggre-gate levels, to guide their judgments and dating business cycles as they occur[NBER (1992)]. These dates are announced with a lag to ensure that thedata on which they are based are as accurate as possible” (Stock and Watson,1999, p.8)

Methods to isolate the cyclical component of time series

There are statistical techniques which permit the researcher to distinguish between thetrend and the cyclical components of economic time series (e.g. linear or nonlinear filters).This separation, however, is not seen as desirable by the proponents of macroeconomicmodels (cfr. real business cycle models), in which the source of long-run growth andshort-run fluctuations is the same (e.g. productivity shocks). Nelson and Plosser (1982)shows that GDP contains a unit autoregressive root, so that is best modelled as a differencestationary (instead of a trend stationary) process.

Methods to analyse co-movements

• Cointegration

• Common factors

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1.3 What causes economic fluctuations?

The causes of macroeconomic fluctuations have been hotly debated by the competingschools of thought. As argued by Romer (2012, pp. 190-191) “the prevailing view isthat the economy is perturbed by disturbances of various types and sizes at more or lessrandom intervals, and that those disturbances then propagate through the economy. Wherethe major macroeconomic schools of thought differ is in their hypotheses concerning theseshocks and propagation mechanisms.”

Competing theories of fluctuations have placed their emphasis on different issues (cfr.Hoover, 2012, pp. 152 -153):

• The propagation mechanism is intrinsically cyclical. Cfr. R. Frisch’s “rockinghorse” model (Frisch, 1933). The argument is that there are economic behavioursthat, although complex, are intrinsically cyclical.

• There are cycles in the impulse mechanism. Cfr. W.S. Jevons (1835-1882) sunspotmodel. One can argue that there are cycles in the agricultural harvest, in technolog-ical change, or in the behaviour of policymakers.

• Instead of putting emphasis on regular behaviour, another class of model put em-phasis on the randomness of both the impulse and propagation mechanism. As Nel-son and Plosser (1982) pointed out, economic output describes a pattern reducibleto a random walk with drift: yt = a+ yt−1 + εt.

Theories of fluctuations can be classified in the following way:

1. Equilibrium theories of fluctuations (Walrasian models), in which the propagationmechanism is characterised as an economic system in which there are no externali-ties, asymmetric information, missing markets, or other imperfections. Everythingis coordinated by the market. Alternative impulse mechanisms have been hypothe-sised:

• Unanticipated monetary shocks in Lucas (1975,1977). These are the so-called“monetary surprise” or “monetary equilibrium business cycle” (MEBC) mod-els.• Real shocks to technology (Real Business Cycle models): Kydland and Prescott

(1982), Long and Plosser (1983).• Incomplete nominal adjustment (New Keynesian models).• Interest rate mismatch: Austrian theory of trade cycle (Hayek 1933).

2. Disequilibrium theories fluctuations (non-Walrasian models of propagation).

• Keynes’s (1936) perspective: rigidities and frictions in wage and price, unsta-ble demand and investment.• Technical change (Schumpeterian models).

3. Marshallian models of fluctuations:

• Monetary disturbances (Friedman-Schwarz 1963)

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2 Real-Business-Cycle Theory

2.1 Tenets of the RBC school

Main features of the Real Business Cycle approach:

1. Emphasis on models as representations of artificial economies:

“One of the functions of theoretical economics is to provide fullyarticulated, artificial economic systems that can serve as laboratoriesin which policies that would be prohibitively expensive to experimentwithin actual economies can be tested at much lower cost” (Lucas, 1980,p.271)

These representations are extremely idealised, without much concern to realism:

“...insistence on the ‘realism’ of an economic model subverts itspotential usefulness in thinking about reality. Any model that is wellenough articulated to give clear answers to the questions we put to it willnecessarily be artificial, abstract, patently unreal” (Lucas, 1980, p.271)

Moreover, they are quantitative representations:

Our task...is to write a FORTRAN program that will accept specificeconomic policy as ‘input’ and will generate as ‘output’ statistics de-scribing the operating characteristics of time series we care about, whichare predicted to result from these policies.

2. Representative agent (household/firm) assumption: it is assumed that the economyis populated by identical individuals. This assumption goes hand in hand with theneoclassical precept of explaining macroeconomic behaviour as the outcome of in-dividual rationality. As Hoover (1995, p. 38) pointed out “[T]he difficulty with thisapproach is that there are millions of people in the economy and it is not practical —nor is it likely to become practical — to model each of them.” To make modellingof rational behaviour feasible new classical macroeconomics9 adopt “representa-tive agent models, in which one agent or a few types of agents stands in for thebehaviour of all agents.”10

9The real business cycle school can be considered as a development of new classical macroeconomics.Snowdon and Vane (2005, p. 294) refers to the RBC school as “New Classical Macroeconomics Mark II”

10For a criticism, cfr. Hoover (1995, pp. 39-40): “using the representative-agent model ... begs thequestion by assuming that aggregation [and interaction] does not fundamentally alter the structure of theaggregate model. Physics may provide a useful analogy. The laws that relate pressure, temperature, andvolumes of gases are macro-physics. The ‘ideal-gas laws’ can be derived from a micromodel: gas moleculesare assumed to be point masses, subject to conservation of momentum, with a distribution of velocities. Anaggregate assumption is also needed: the probability of the gas molecules moving in any direction is takento be equal. ... Unlike gases, human society does not comprise homogeneous molecules, but rational people,each choosing constantly. To understand (verstehen) their behaviour, one must model the individual and hissituation. This insight is clearly correct, it is not clear in the least that it is adequately captured in the heroic

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3. Maximization: households/firms (the representative household/firm) maximize theirutility/profits under constraints. The output produced by firm is determined wheremarginal revenues equal marginal costs.

4. Rational expectations hypothesis: agents (the representative agent) form their ex-pectations rationally. As Hoover (1988, pp.14-15) points out, there can be threeinterpretations of the REH. A weak form is that “people learn from their mistakesand do no persist in them.” This raises the question of how people learn, but newclassical macroeconomics does not address this question. A strong form is that“people actually know the structure of the model that truly describes the world anduse it to form their expectations.” This would imply that economic agents applyan incredible cognitive power before taking any economic decision. An alternativeinterpretation goes back to the original formulation of Muth (1961) and refers to therelationship between the predictions one can draw from the model and the predic-tions that the agents within the model can make: the predictions that the modelledagents are able to do are not worst than the predictions one can imply from themodel (cfr. Muth, 1961). This view leaves open the question of how models relateto the real world (cfr. Hoover, 1988, p.15). New classical macroeconomics tends toendorse the second or the third interpretation.

5. Continuous equilibrium: price flexibility ensures continuous market clearing, thereis no moment in which the labour and goods markets are not in equilibrium. In eachmarket the amount of goods and services demanded is always equal to the amountproduced or made available. As demand and/or supply change, prices change im-mediately. This assumption is shared with new classical macroeconomic modelMark I (Lucas Jr, 1972, cfr.). In this setting economic agents (workers, consumersand firms) are price takers (i.e. they do not have market power to influence price):‘perfect competition’ reigns. There are no externalities, thus the equilibrium isPareto-optimal. Unemployment is a voluntary phenomenon (Lucas, 1978). Theassumption of continuous equilibrium (together with the REH) is the most criti-cal and controversial assumption. It is not, of course, shared with Keynesian ap-proaches, but even the monetarist school of Friedman allowed the possibility ofdisequilibrium in the short-run. The reduction of macroeconomics to Walrasiangeneral equilibrium microeconomics has been referred to by Hoover (1988, p.87)as “the euthanasia of macroeconomics.”

6. The dominant impulse mechanism is represented by random changes in the avail-able production technologies (exogenous shocks to technology). Some RBC mod-els include also shocks to government purchases.

7. The Propagation mechanism is represented by:

aggregation assumptions of the representative-agent model. The analogue in physics would be to model thebehaviour of gases at the macrophysical level, not as derived from the aggregation of molecules of randomlydistributed momenta, but as a single molecule scaled up to observable volume—a thing corresponding tonothing ever known to nature.” On a similar spirit Kirman (1992) severely criticizes the representative-agent assumption because it fails to fulfil the necessary conditions for perfect aggregation. In this waythe representative agent does not faithful represent the actual individuals, even if their behaviour satisfiesrational choice theory.

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• consumption smoothing

• lags in the investment process (time to build )

• intertemporal labour substitution

8. Fluctuations in employment mainly voluntary due to the substitutability of workand leisure.

9. Downplaying monetary policy (money is of little importance in business cycles),neutrality of money.

10. Breaking down of the cycle/trend (short-run/long-run) dichotomy. RBC theoristsclaim that the same theory that explains long-run growth should also explain short-run fluctuations.

11. Calibration. Calibration is a strategy for finding numerical values for the param-eters of the highly idealised models that the RBC school proposes. It consists inboth (i) selecting, through simulation, values for parameters so that the model iscapable of reproducing some statistical properties of the observed economic timeseries, and (ii) choosing the parameters of the model from values picked from pre-existing microeconomic studies or from general facts about national-income ac-counting. Calibration eschews standard econometric testing and is also alternativeto the conventional econometric methodology consisting in adequately specifying amodel, finding out the reduced-form, and directly estimating it from the data usingregression techniques. As Hoover (1995, p.29) puts it “[t]he dominance of theoryin the choice of models lies at the heart of the difference between estimators andcalibrators.” With estimation one can compare models from alternative theories tosee which one is more consistent with the data. With calibration “the aim is neverto test and possibly reject the core theory, but to construct models that reproducethe economy more and more closely within the strict limits of the basic theory....thereal-business-cycle modeller typically does not regard the core theory at risk inprinciple. Like the estimators, the calibrators wish to have a close fit between theirquantified model and the actual data —at least in selected dimensions. But the fail-ure to obtain a close fit (statistical rejection) does not provide grounds for rejectingthe fundamental underlying theory (Hoover, 1995, p.29).” In conclusion “[t]he cal-ibration methodology..lacks any discipline as stern as that imposed by econometricmethod (Hoover, 1995, p.41).”

2.2 Basic framework

The basic framework for RBC analysis is the neoclassical model of capital accumulation(cfr. Solow 1956, 1957), to which RBC theorists will add shocks to productivity.

The Solow model focuses on four variable: output (Y), capital (K), labour (L) and“knowledge” or “technology” (A). It postulates a one-good economy, in which the capitalstock is just the accumulation of this composite commodity. The production functiontakes the form:

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Y (t) = F (K(t), A(t)L(t)), (1)

where t denotes time (cfr. Romer 2001, ch. 1.2). AL is referred to as effective labourand technological progress is said to be labour-augmenting or Harrod-neutral. It is as-sumed constant return to scale (i.e. the production function is homogeneous of degreeone: F (λK, λAL) = λF (K,AL)) and that the structure of the market is perfectly com-petitive, so that real wage is equal to the marginal product of labour (mpl) and real interestrate is equal to the marginal product of capital (mpk).

A Cobb-Douglas production function is usually assumed:

Y = F (K,AL) = Kα(AL)1−α, 0 < α < 1 (2)

which has the following specificities (that make the model easy to analyse): (i) constantreturn to scale; (ii) Y increases with each factor of production ceteris paribus; (iii) theproduction function goes to the origin (no free lunch); (iv) diminishing returns to eachfactor of production; (v) mpl raises if K increases ceteris paribus, mpk raises if L in-creases ceteris paribus, and both mpk and mpl raise if A increases ceteris paribus 11; (vi)if real wage are equal to mpl, then the labour share in output, defined as real wages×L

Y=

mpl × LKα(AL)1−α

= 1 − α. Analogously, if the real interest rate is equal to mpk then thecapital share in output, defined as real interest rate ×K

Y= mpk × K

Kα(AL)1−α= α. That

is, L−share and K−share are constant. Notice that they remain constant even if the CobbDouglas is formulated such that technology is capital augmenting or Hicks-neutral. In-deed there are no difference among these three formulations as far as the Cobb-Douglasis concerned.

Equation (2) can be written as:

log Y = α logK + (1− α) logL+ (1− α) logA (3)

From this equation, taking derivatives with respect of time, one can get:

gy = αgk + (1− α)gl + z, (4)

where gy, gk, gl are the growth rates of output, capital and labour and z = (1 − α)d logAdt

measures the growth in output that cannot be accounted for by growth in capital andlabour. Thus z represents total factor productivity growth and has been referred to as the“Solow residual,” since equations (3) and (4) have been empirically estimated throughOLS regression. Equation (4) can be rewritten as:

gy + (1− α)gy − (1− α)gy = αgk + (1− α)gl + z (5)

(gy − gl) =

(α

1− α

)(gk − gy) +

(1

1− α

)z (6)

Thus, growth of output per capita depends on the growth of the capital-output ratio

11mpl = ∂Y∂L = (1− α)KαA1−αL−α; mpk = ∂Y

∂K = αKα−1(AL)1−α

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and on the Solow residual. The Solow residual has accounted for approximately half thegrowth in output in the U.S. between 1870s and 1980s (cfr. Blanchard and Fisher 1989 andStadler 1994). This residual has been observed fluctuating significantly over time and hasbeen described as a random walk with drift plus some serially uncorrelated measurementerrors (Prescott 1986, Stadler 1994). RBC theorists incorporate stochastic fluctuationsin the rate of technical progress into the neoclassical growth model such that this candisplay business cycle phenomena. “Thus, RBC theory can be seen as a development ofthe neoclassical growth theory of the 1950s” (Stadler, 1994, p.1753).

2.3 The basic RBC model

We consider now a basic RBC model, as described by Romer (2001, ch. 4.3) [Romer(2012, ch. 5.3)].

• Assumptions: identical price-taking firms/ (infinitely lived) households

• Production (inputs: capital K, labour L, and ‘technology’ A):

Yt = Kαt (AtLt)

1−α, 0 < α < 1 (Cobb-Douglas prod. fun.) (7)

• Capital (N.B. Y ≡ C + I +G):

Kt+1 = Kt + It − δKt = Kt + Yt − Ct −Gt − δKt (8)

where C is consumption, I investment, G government purchases, δ is the rate ofdepreciation of capital.

• L and K are paid their marginal products. To the m.p.k. it has to be subtracted thedepreciation rate δ. Thus real wages (w) and real interest rate (r) are:

wt =∂Yt∂Lt

= (1− α)

(Kt

AtLt

)αAt (9)

rt =∂Yt∂Kt

− δ = α

(AtLtKt

)(1−α)

− δ (10)

• Representative household optimization problem: max expected value of

U =∞∑t=0

e−ρtu(ct, 1− lt)Nt

H(11)

where:

– u(·): instantaneous utility function of the representative member of the house-hold;

– ρ: discount rate;

16

– Nt: population; H: number of households; Nt/H: number of membersof the household;

– c: consumption per member, 1− l: leisure per member; l: amount each mem-ber works;

– c ≡ C/N ; l ≡ L/N .

• Population growth:logNt = N + nt n < ρ (12)

• Log-linear u(·)ut = log ct + b log(1− lt) b > 0 (13)

• Technology path:logAt = A+ gt+ At (14)

At = ρAAt−1 + εA,t, −1 < ρA < 1,where εA,t’s are white-noise disturbances: a series of mean-zero and equal varianceshocks that are uncorrelated with one another.

• Government purchases :

logGt = G+ (n+ g)t+ Gt (15)

Gt = ρGGt−1 + εG,t, −1 < ρG < 1,where εG,t’s are white-noise disturbances.

2.4 Household behaviour

We examine now the consequence that the inclusion of leisure in the utility function andthe introduction of randomness in technology and government purchases have for house-holds’ behaviour. We closely follow the analysis of Romer (2001, ch. 4.4) [Romer (2012,ch. 5.4)].

Special case 1: one period life

Consider the special case in which household lives only for one period, has no initialwealth, and has only one member. In this case U = log c + b log(1 − l). Household’sbudget constraint is c = wl (individual consumption is equal to income and there is nosaving since there is only one period). The Lagrangian for the household’s maximizationproblem is

L = log c+ b log(1− l) + λ(wl − c). (16)

The first order conditions are:

∂L∂c

=1

c− λ = 0 ⇒ λ =

1

c

17

∂L∂c

= − b

1− l+ λw = 0 ⇒ (replacing λ with

1

c) − b

1− l+

1

cw = 0

Replacing w with cl

we get:

l =1

b+ 1(17)

Using the budget constraint equation we get:

c =w

b+ 1(18)

Notice that in this particular case labour supply does not depend on wage. Generallyspeaking, real wages have two counteracting effects on household’s decision to supplylabour and to consume. The first effect is the income (or wealth) effect: higher realwages make people feel wealthier, this will tend to suppress the supply of labour (ce-teris paribus) and will tend to increase present or future consumption (ceteris paribus).The second effect is the substitution effect: higher real wages make people feel prof-itable to replace leisure with work and get an extra income for present or future consump-tion. There is also a substitution effect on consumption: consumption can be postponed(present consumption replaced with future consumption) depending on the interest rate(the higher the interest rate, the more profitable is to save and procrastinate consumption).

In this first simple one-period case, as regards the supply of labour income and sub-stitution effects counterbalance each other, so that the supply of labour is constant. Asregards consumption, there cannot be a substitution effect since the household liven onlyone period.

Special case 2: two-periods life

Consider the same case as before except that households live two periods. There is nouncertainty about the (second-period) interest rate or the second-period wage. In thiscase U = log c1 + b log(1− l1) + e−ρ[log c2 + b log(1− l2)] and the budget constraint isc1 + c2 = w1l1 + r(w1l1 − c1) + w2l2 (being the return on saving an additional source ofincome in the second period). The Lagrangian for the household’s maximization problemis now

L = log c1+b log(1−l1)+e−ρ[log c2+b log(1−l2)]+λ[w1l1+r(w1l1−c1)+w2l2−c1−c2](19)

The first order conditions are:

∂L∂l1

= − b

1− l1+ λw1(1 + r) = 0 ⇒ λ =

b

1− l11

w1(1 + r)

∂L∂l2

= − b e−ρ

1− l2+ λw2 = 0 ⇒ λ =

b e−ρ

1− l21

w2

18

∂L∂c1

=1

c1

− λ(1 + r) = 0 ⇒ λ =1

c1(1 + r)

∂L∂c2

=e−ρ

c2

− λ = 0 ⇒ λ =e−ρ

c2

We get:1− l11− l2

=1

e−ρ(1 + r)

w2

w1

(20)

which describes how relative leisure responds to relative wage and real interest rate, and

c1

c2

=1

e−ρ(1 + r)(21)

which describes how relative consumption responds to the real interest rate.

Equation (20) implies that an exogenous shock which causes present real wage tobe higher relative to future real wage will increase present labour supply relatively tofuture labour supply (increase future leisure relatively to present leisure). A rise in rincreases the attractiveness of working today (and saving) relative to working tomorrow.These responses of labour supply to the relative wages and the interest rate are known asintertemporal substitution in labour supply and have been analysed by Lucas and Rapping(1969) (cfr. Romer, 2001, p.178).

From equation (20) it follows that the elasticity of substitution between leisure in the

two periods(d log

(1−l11−l2

)d log

(w2w1

))

is equal to one. As regard the relative consumption, equation

(21) shows that in this particular case it responds to interest rate only.

Household optimization in the general case

Let us now turn to the general case, in which there is uncertainty about future r and w.Because of shocks to technology (and government purchases), the representative may con-sider to reduce current consumption per member by a certain amount, save that portion ofcurrent income, and then use the resulting extra-income (obtained from return to saving)to increase consumption per member in the net period. If household behaves maximally,the household chooses c such that any small increase ∆c leaves the marginal expectedbenefit equal to the marginal disutility (cost) of decreasing consumption at time t.

Recall that population grows at rate n:

logNt = N + nt,

where N is a constant. It follows that Nt+1 = enNt. This can be shown recursively,starting from period 0:

19

N1 = enN0

log(N1) = n+ log(N0)

log(N2) = n+ log(N1) = n+ n+ log(N0)

log(N3) = n+ log(N2) = 3n+ log(N0)

log(Nt) = nt+ log(N0) = N + nt,

where log(N0) = N .

Marginal utility of consumption (m.u.c.) in period t per member (see again equations11 and 13 ) is:

∂U

∂ct= e−ρt

1

ct

Nt

H(22)

Thus the utility cost in decreasing consumption by ∆c is e−ρt 1ctNtH

∆c.

The increase in consumption in the next period is equal to ∆c plus the return to saving,divided by the amount by which population has grown, that is: ∆c+rt+1∆c

en.

The expected utility benefit is equal to m.u.c. at period t+1 per member(e−ρ(t+1) 1

ct+1

Nt+1

H

)times the increase in consumption in period t+ 1:

Et

[e−ρ(t+1) 1

ct+1

Nt+1

H

∆c(1 + rt+1)

en

]Since Nt+1 = enNt and e−ρ(t+1), Nt+1

H, ∆c are not uncertain (the only unknowns are ct+1

and rt+1), the latter expression simplifies to:

Et

[1

ct+1

(1 + rt+1)

]e−ρ(t+1)Nt

H∆c

By equating utility cost(e−ρt 1

ctNtH

∆c)

to expected benefit we get:

1

ct= e−ρEt

[1

ct+1

(1 + rt+1)

]= e−ρ

{Et

[1

ct+1

]Et [1 + rt+1] + Cov

(1

ct+1

, 1 + rt+1

)}.

(23)This means that there is a trade-off between present and future consumption. How muchis consumed today (ct) depends on the expectation of how much is consumed tomorrow(this in turn depends on the expectation of the m.u.c. at period t + 1), on the expectationof the rate of return, and on the interaction of these two variables. Suppose, for example,ct+1 is positively correlated to rt+1. This means that Cov

(1

ct+1, 1 + rt+1

)< 0. In this

case, ceteris paribus, ct will be higher (and saving lower) with respect to the case in whichct+1 and rt+1 are uncorrelated. If c and r are uncorrelated ore negatively correlated (thislatter case would imply correlation between saving and r), then saving today respondspositively to rates of return of tomorrow.

20

Trade-off between consumption and labour supply

The more a household works, the higher is its income which can be allocated to consump-tion. But the more a household works, the less is the leisure it can enjoy. Thus there is atrade-off between (present) consumption and (present) labour supply. Consider the rep-resentative household increasing its labour supply per member by a small amount ∆l toincrease consumption. Again, if the household is behaving optimally the marginal disutil-ity of giving up leisure should be equal to the marginal utility of increasing conusmption.That is

e−ρtNt

H

b

1− lt∆l = e−ρt

Nt

H

1

ctwt∆l, (24)

which reduces toct

1− lt=wtb

(25)

Notice that in this general case, both ct and lt responds positively to wt.

2.5 Solving the model in a special case

Two simplifying assumptions (cfr. Long and Plosser, 1983; Romer, 2001):

• no government

• δ = 1, i.e. 100% depreciation each period

These assumptions imply

Kt+1 = It +Kt − δKt = It = Yt − Ct (26)

1 + rt = α

(AtLtKt

)1−α

(27)

As Romer (2001, p.181) puts it: “The elimination of government can be justifiedon the grounds that doing so allows us to isolate the effects of technology shocks. Thegrounds for the assumption of complete depreciation, on the other hand, are only that itallows us to solve the model analytically.”

Let us call st the fraction of output which is saved, i.e. st = 1 − CtYt

. It follows thatct = (1− st) YtNt .

Consider again equation (23): 1ct

= e−ρEt

[1+rt+1

ct+1

]. Replacing ct with (1− st) YtNt and

taking logs we get:

− log

[(1− st)

YtNt

]= −ρ+ logEt

[(1 + rt+1)Nt+1

(1− st+1)Yt+1

]. (28)

21

Since 1 + rt+1 = mpk = α(At+1Lt+1

Kt+1

)1−α= αYt+1

Kt+1and Kt+1 = It = Yt − Ct = stYt, we

get:

log st − log(1− st) = −ρ+ n+ logα + logEt

[1

1− st+1

](29)

If st+1 is constant over time, this constant s is a solution for the equilibrium condition(which can be proved to be unique):

s∗ = αen−ρ (30)

Thus in equilibrium, which is also the situation the representative household maximizesthe expected utility, under the two simplifying assumption stated above, the saving rate isconstant.

Consider now equation (25), the other key equation describing household’s optimizingbehaviour: ct

1−lt = wtb

. We have:

log

[(1− s∗) Yt

Nt

]− log(1− lt) = logwt − log b (31)

Recall that real wage is equal to the marginal productivity of labour (see equation 9).Since the production function is Cobb-Douglas Mpl = (1 − α)Kα

t A1−αt L−αt = (1 − α) Yt

Lt

= (1− α) YtltNt

.

This yields:

log(1−s∗)+log Yt−logNt−log(1−lt) = log(1−α)+log Yt−log lt−logNt−log b (32)

After some algebraic manipulation, we get:

lt =1− α

(1− α) + b(1− s∗)≡ l∗. (33)

Thus optimal labour supply (under the two simplifying assumptions) is also constant. AsRomer (2001: p. 183) puts it: “The reason this occurs despite households’ willingnessto substitute their labour supply intertemporally is that movements in either technologyor capital have offsetting impacts on the relative-wage and interest-rate effects on laboursupply.”

Very schematically:

↑ At =⇒↑ wtEt[wt+1]

=⇒↑ lt

But also↑ At =⇒↑ s Yt

Nt

=⇒↓ Et[rt+1] =⇒↓ lt

22

Output fluctuations as AR(2)

Recall (we continue to assume G = 0 and δ = 1):

Yt = Kαt (AtLt)

1−α Kt = s∗Yt−1 logAt = A+ gt+ At logNt = N + nt

We get:

log Yt = α log s∗ + α log Yt−1 + (1− α)(A+ gt+ At + log l∗ +N + nt)

Call Yt: deviations of log Yt from the normal path (difference between log Yt and value itwould take if At = 0)

Yt = αYt−1 + (1− α)At (34)

Since At = ρAAt−1 + εA,t:

Yt = (α + ρA)Yt−1 − αρAYt−2 + (1− α)εA,t AR(2) process (35)

This process is able to create hump-shaped responses to the shock εA,t (cfr. Romer 2001,pp. 184-185; Romer 2012, pp.205-206).

3 Keynesian Theories of Fluctuations

3.1 The IS/LM model

The IS/LM model was introduced by John Hicks in an article published in 1937 in Econo-metrica with the title “Mr Keynes and the Classics: A Suggested Reinterpretation.”

It summarizes in a schematic manner the mechanism by which the level of output isdetermined by aggregate demand in the short run, i.e. the period in which wages andprices are sticky. Wages and prices do not respond immediately to changes in demand,because, for example, institutional arrangements (so that e.g. wages are reviewed period-ically and not continuously) or menu-costs (i.e. there are costs associated with changingprices).

The model consists in two parts: the goods market and the money market. The goodsmarket is described by the IS curve, which denotes a situation in which there is a short-runequilibrium. IS stands for “Investment-Savings”: in equilibrium planned investment mustbe equal to planned savings. The equilibrium in the money market is described by the LMcurve. LM stands for “Liquidity-Money”: in equilibrium money demand, i.e. demand forliquidity, must be equal to money supply. (Cfr. Carlin and Soskice, 2006, pp.28-29).

3.1.1 The goods market: the IS curve

“The IS curve shows the combinations of output and the interest rate such that plannedand actual expenditures on output are equal” Romer (2001, p.219). Under the assumption

23

that wages and prices are fixed, this means that there is equilibrium in the goods market:aggregate demand equals supply.

Let us denote planned real expenditure by Y D and real output by Y . We have

Y D = C(Y, T,W ) + I(r, A) +G, (36)

where

• C(Y, T,W ) is consumption (by households) as a function of real output Y , totaltaxation T , and wealth W . Using a linear consumption function we have:

C(·) = c0 + cy(1− ty)Y,

where c0 is autonomous consumption, which includes factors such as wealth andexpected future income, and (1 − ty)Y is disposable income (ty is the portion ofincome that is taxed, so that we assume a linear tax function and 0 < ty < 1).The term cy (0 < cy < 1) is the marginal propensity to consume out of disposableincome.

• I(r, A) is investment (by firms) as function of real rate of interest r (r influencesnegatively investment) and a term A which captures expected future profitability.12

“The simple idea is that firms are faced with an array of investment projects, whichare ranked by their expected return. If the interest rate falls, then this reduces thecost of capital and makes some projects profitable that would not otherwise havebeen undertaken” (Carlin and Soskice, 2006, p.30). Using a linear investment func-tion:

I = A− ar,

where a is a constant.

Thus we have:Y D = c0 + cy(1− ty)Y + A− ar +G. (37)

Equilibrium condition:

Y =1

1− cy(1− ty)[c0 + (A− ar) +G] (38)

Let sy = (1− cy) be the marginal propensity to save. We get:

Y =1

sy + cyty[c0 + (A− ar) +G] (39)

The term 1sy+cyty

is called the multiplier. The IS curve is derived graphically in Figure(4).

12Actually interest rate is likely to influence consumption of durable goods as well and investment islikely to be influenced by income. For a more general formulation see Romer (2001, p.220).

24

Figure 4: Investment curve and IS curve. Source: Carlin and Soskice (2006, Figure 2.1).g denotes government purchases (denoted by G in the text of this note).

Notice that for r = 0, Y = c0+A+Gsy+cyty

(intercept of the IS curve on the Y -axis). ForY = 0, r = c0+A+G

a(intercept of the IS curve on the r-axis). The IS curve as a function

of r is indeedr =

c0 + A+G

a− sy + cyty

aY.

It is easy to see that (cfr. Carlin and Soskice, 2006, pp. 32 and 63-64):

1. If the multiplier(

1sy+cyty

)changes (e.g. cy rises), then it changes the intercept

of the IS curve on the Y-axis, but not the intercept on the r-axis. A rise in themultiplier will make the IS flatter: it rotates counter-clockwise from the intercepton the r-axis.

2. Conversely, a change in a (interest-sensitivity to investment) changes the intercepton the r-axis, but not the intercept on the Y -axis. Thus, a fall in a rotates the curveclockwise from the intercept on the Y axis.

3. Any change in c0, G, and A will cause the IS curve to shift (by the change times themultiplier) leaving the slope unchanged.

3.1.2 The money market: the LM curve

“The LM curve shows the combinations of output and the interest rate that lead to equilib-rium in the money market for a given price level” (Romer, 2001, p.222). In the followingdiscussion it is also assumed that the central bank controls directly the money supply.

25

Demand for money:

MD

P= L(y, i),

∂L

∂i< 0,

∂L

∂Y> 0 (40)

The demand for money is a decision about the form in which to hold wealth. Wealthis the set of valuable things, i.e. assets, that you own. While income is a flow, wealth isa stock. Assets can be monetary or nonmonetary. Which kind of assets should you own?You may own cash, money in a bank accounts where you can have immediate access bywriting a cheque, using a debit card or withdrawing money by an ATM machine, you mayhave money in a bank account which pays you interest rates or you may own bonds whichalso pay interest. Bonds offer you an interest as long as you are prepared to take somerisk.

The allocation of wealth between interest-bearing and not interest-bearing assets willdepend on the interest rate (jointly with your perception of risk and willingness to takesome risk) and the volume of transactions in the economy, which in turns depends on thelevel of income. Indeed to carry out transactions you need money. But holding moneybears an opportunity cost: you give up the interest income you have got holding bonds(or other interest-bearing assets). There is then a trade-off. A rise in the interest rate willshift the balance in favour of the demand for not interest-bearing assets. Thus the demandfor money depends positively on income

(∂L∂Y

> 0)

and negatively on the nominal interestrate

(∂L∂i< 0).

Following Carlin and Soskice (2006, p.35), we have

MD

P= L(Y, i) = l − lii+

Y

vT, (41)

where l − lii represents asset demand (i.e. the asset motive for holding money) and YvT

represents transactions demand (i.e. the transactions motive for holding money). Theterms l, li, and vT are here considered constants.

Notice that:

• In the Quantity Theory of Money (well established proposition in classical eco-nomics since the XIX century) we just have MD

P= Y

vT. vT is the velocity of circu-

lation of transactions, sometimes simply referred to as the “velocity of money”.

• The allocation of wealth between money and interest-bearing assets is also drivenby speculative motives, as underlined by Keynes (1936). Tobin (1958) analysed theinteraction between liquidity preference and behaviour toward risk.

In equilibrium we have that money demand is equal to money supply:

MD

P=MS

P(42)

L(Y, i) =MS

P(43)

26

Figure 5: LM curve. Source: Carlin and Soskice (2006, Figure 2.4). Excess supply leadsto a fall in i, excess demand leads to a rise in i.

In the standard IS/LM model it is assumed that MS is fixed by the monetary authority.

As the IS curve, we draw the LM curve in the i × Y plan. We need to express i as afunction of Y . We get:

i =1

li

(l − MS

P

)+

1

li

1

vTY (44)

Thus, when Y = 0, i = 1li

(l − MS

P

), when i = 0, Y = vT

(MS

P− l)

.

We have that (cfr. Carlin and Soskice, 2006, pp. 38-39,65):

• A rise in vT , the transactions velocity of circulation, moves the intercept of the LMcurve with the Y−axis to the right (while the intercept with the i−axis does notchange): it rotates the LM curve clockwise making it flatter.

• A rise in li, the interest sensitivity of the demand for money, will shift the interceptwith the i−axis towards the origin, while the intercept with the Y−axis does notchange: it rotates the LM curve clockwise making it flatter.

• A change in the money supply will shift the LM curve rightwards by vT (∆MS/P )(no change in the slope).

• A change in the price level will shift the LM leftwards by vT (MS/∆P ) (no changein the slope).

27

3.1.3 The relationship between real and nominal interest rates

The relationship between the real and the nominal interest rate is

1 + r = (1 + i)P

PEt+1

, (45)

where r is the real interest rate, i is the nominal interest rate, P is the price level and PEt+1

is the expected price level in the future. Let expected inflation πE be equal to PEt+1−PP

. Wehave that PπE = PE

t+1 − P . Then PπE

PEt+1= 1− P

PEt+1. Hence

P

PEt+1

=1

1 + πE

1 + r =1 + i

1 + πE

r =i− πE

1 + πE(46)

If πE is low the relationship between i and r is approximated as

i ' r + πE (47)

Notice that i can be observed, but r (and πE) only estimated.

3.1.4 The interplay between the IS and LM curve

The IS/LM scheme is useful to study the effect of fiscal policy (Figure 6-a) and the ef-fect of monetary policy (Figure 6-b). As the case of the liquidity trap shows (Figure 7)monetary policy may be insufficient to generate new output.

28

Figure 6: Comparative Statics in the IS/LM model Source: Carlin and Soskice (2006,Figure 2.5). Chart (a): the effects of an increase in government purchases. Chart (b): theeffects of a rise in money supply

29

Figure 7: The liquidity trap. Source: Carlin and Soskice (2006, Figure 2.6).

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