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T HEORIES OF F LUCTUATIONS
Lecture notes prepared for the course:
Advanced Macroeconomics (mod.2)
Master of Science in Economics
University of Pisa - Scuola Superiore Sant’Anna
Alessio M ONETA1
April 29, 2015
Contents
1
2
3
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
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
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 . . . . . . . . . . . .
1
28
Institute of Economics, Scuola Superiore Sant’Anna, Piazza Martiri della Libertà 33, 56127 Pisa, Italy.
Email: [email protected]. The author is very grateful to be informed of any mistake found in this text!
2
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 tasks
of macroeconomics. According to Snowdon and Vane (2005, p. 304), for example, the
central goal of macroeconomics is to provide “coherent and robust explanations of aggregate movements of output, employment and the price level, in both the short run and the
long run”. But before examining the causes of these phenomena, we have to identify their
typical features and find out how they can be quantitatively described.
When we talk about economic fluctuations or business cycles2 the focus is usually
on the short run (few months) or middle run (few years).The focus on the short run was
particularly emphasized by J.M. Keynes: “In the long run we are all dead. Economists
set themselves too easy, too useless a task if in tempestuous seasons they can only tell us
that when the storm is long past the ocean is flat again” (Keynes, 1923). When Keynes
wrote these lines, his polemical target was the quantity theory of money and the connected
principle that money is neutral3 in the long run. Classical economists (e.g. Ricardo) were
concerned with long-run equilibrium. Keynes’s analysis, which culminated with the publication of the General Theory (Keynes, 1936), shifted the emphasis to the short-run, and
to out-of-equilibrium states of the economy. In his work Keynes aimed to explain the
causes of economic instability since he argued that capitalist market economies can stay
at less than full employment for prolonged period of time, and the Great Depression witnessed that this was indeed possible. However, it would be incorrect to say that Keynes
was just interested in the short run. The importance of long-term consequences of economic decisions was duly recognized by Keynes (cfr. Keynes, 1930). But it is true that its
emphasis on instability and out-of-equilibrium phenomena reconsidered the importance
of a careful analysis of the short-run fluctuations.
While Keynes advocated to enrich economic theory with the analysis of economic
fluctuations, at the same time other, less theoretical, approaches emerged. In the U.S. the
National Bureau of Economic Research (a private nonprofit research organization) was
founded in 1920. The first director of research was Wesley Mitchell who pioneered empirical research of business cycle phenomena. In 1946 Arthur Burns (chair of the Federal
Reserve form 1970 to 1978) and Wesley Mitchell (an important figure in the American
institutionalism) published a book (Measuring Business Cycles), which contains the following definition of business cycles:
2
Economic fluctuations or business cycles? Which is the right term and what is the difference among
them? They are synonymous. Economists use these two terms interchangeably. Those who want to emphasize the regular feature of the ups and downs of the economy use the term “business cycle.” The term
“economic fluctuations” is perhaps more neutral.
3
Money is neutral if after a monetary expansion (increase in the supply of money) the price level, nominal 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 economic activity of nations that organise their work mainly in business enterprises. A cycle consists of expansions occurring at about the same time in
many economic activities, followed by similarly general recessions, contractions, and revivals which merge into the expansion phase of the next cycle:
this sequence of changes is recurrent but not periodic, in duration business
cycles vary from more than one year to ten or twelve years.” (Burns and
Mitchell, 1946)
The main features of a business cycle are therefore: (i) alternation in the state (ups
and downs) of the economy; (ii) a recurrence of the ups and downs, although not perfectly 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 1947
to 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 that
the recurrence of ups and downs does not display a regular cycle, but the alternation does
not appear to be completely random either. Before looking at some stylized facts about
economic 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).
4
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 recessions. The dates are usually announced with a certain lag. Its definition of recession
is “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.” Notice that the NBER’s decision is not the result of
any formal algorithm but is ultimately based on overall impressions on the movements of
many economic indicators.4
In Europe the Centre for Economic Policy Research, a think tank based in UK, defines
European recession as “a significant decline in the level of economic activity, spread
across the economy of the euro area, usually visible in two or more consecutive quarters of
negative growth in GDP, employment and other measures of aggregate economic activity
for 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 duration), e.g. Great Depression of 1929-19336 ; 2007-2009 US Great Recession7 ; Eurozone
crisis 2009-?
Growth recession (slowdown): period of slower than trend growth
Cycle: (i) period between trough and next trough, or (ii) period between peak and
next peak.
Recovery: (i) period from trough to the level of the previous peak, or (ii) period from
trough 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 economic fluctuations.
4
Cfr. http://www.nber.org/cycles.html
Cfr. http://www.cepr.org/data/dating
6
In 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.
7
Technically 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. Between
December 2007 and June 2009 GDP fell by 4.3 percent, the largest fall during a recession since the Great
Depression 1929-1933.
5
5
Figure 2: A stylised economic time series, from Hoover (2012) Applied Intermediate
Macroeconomics, 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 exhibit
any strictly regular cyclical pattern. Figure 1 shows that output declines are of different
size and nature. Table 1 shows that recessions vary in terms of length and changes in
ouput and unemployment.
Table 1: Recessions in the United States since World War II
Recession dates
Nov. 1948-Oct. 1949
July 1953-May 1954
Aug. 1957-Apr. 1958
Apr. 1960-Feb. 1961
Dec. 1969-Nov. 1970
Nov. 1973-Mar. 1975
Jan. 1980-July 1980
July 1981-Nov. 1982
July 1990-Mar. 1991
Mar. 2001-Nov. 2001
Dec. 2007 - June 2009
Number of months
from peak to through
Change in real GDP
from peak to through
Highest
unemployment rate*
11
10
8
10
11
16
6
16
8
8
18
-1.7%
-2.7
-1.2
-1.6
-0.6
-3.1
-2.2
-2.9
-1.3
-0.5
-5.1
7.9%
5.9
7.4
6.9
5.9
8.6
7.8
10.8
6.8
6.0
10
*included the aftermath period of the recession
Source: NBER
6
2. Symmetries and asymmetries in output movements
In the U.S. (post-World War II) output growth is distributed roughly symmetrically
around 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 recession
phase. Figure 3 shows U.S. real GDP growth rates (quarterly compound annual and annual)8 . Notice that mean is positive (around 3.5%).
Real GDP Growth Rates, US 1948−2012
15
quarterly growth rate
(compound annual)
annual growth rate
5
0
−10
−5
percent per year
10
mean a. growth rate
1950
1960
1970
1980
1990
2000
2010
years
Figure 3: U.S. real GDP Growth rates (quarterly and annual)
8
A growth rate is the proportionate (or percentage) rate of change per unit time and is calculated as
X̂t =
Xt − Xt−1
Xt
=
− 1.
Xt−1
Xt−1
If data are observed quarterly, then X̂t is a quarterly growth rate. For the sake of comparisons it is useful to
convert a quarterly growth rate in annual units. We get the compound annual quarterly growth rate:
X̂t =
Xt
Xt−1
4
− 1.
The annual growth rate computed from quarterly data is:
X̂t =
Xt
− 1.
Xt−4
We see from Figure 3 that the annual growth rate is smoother than the compound annual quarterly growth
rate (see Hoover, 2012, pp. 764-766).
In continuous time the growth rate of X(t) corresponds to the derivative of log X(t). Indeed
X̂(t) =
Ẋ(t)
dX(t) 1
d log(X(t))
=
=
.
X(t)
dt X(t)
dt
7
Table 2: Behaviour of the components of output in recessions (U.S. 1947 - 2009)
Components of GDP
Average share
in real GDP
Average share in fall
in GDP in recessions
relative to normal growth
Consumption
Durables
Nondurables
Services
8.9%
20.6
35.2
14.6%
9.7
10.9
Investment
Residential
Fixed nonresidential
Change in inventories
4.7
10.7
0.6
10.5
21.0
44.8
-1
-12.3
Net exports
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 the
average shares of the same components in the declines of output (relative to the normal
growth) 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 relative
to GDP normal growth. In sum, some components of GDP fluctuates much more than
others.
4. Co-movements
Another important feature of the business cycles is the co-movements of many economic variables in a predictable way. This has led Lucas (1977) to claim that “with
respect to the qualitative behaviour of co-movements among series business cycles are all
alike”. It suggests “the possibility of a unified explanation of business cycles grounded in
the general laws governing market economies, rather than in political or institutional characteristics specific to particular countries or periods” (Lucas, 1977, p.10). It is not obvious
to identify general laws in economics, but it is true that there are robust empirical regularities about the deviations from trend of many economic variables. Table 3 summarizes the
typical cyclical behaviour of a wide range of key economic variables. Variables that move
in the same direction (display postive correlation) as GDP are procyclical; variables that
move in the opposite direction (display negative correlation) to GDP are countercyclical;
variables that display no clear pattern (zero correlation) are acyclical. With respect to
timing, variable that move ahead of GDP are leading variables; variables that follow GDP
are lagging variables; and variables that move at the same time as GDP are coincident
variables (Snowdon and Vane, 2005, p. 307).
8
Table 3: Co-movements in business cycle
Variable
Production
Industrial production
Direction
Timing
Procyclical
Coincident
Expenditure
Consumption
Business fixed investment
Residential investment
Inventory investment
Government purchase
Procyclical
Procyclical
Procyclical
Procyclical
Procyclical
Coincident
Coincident
Leading
Leading
-
Labour market variables
Employment
Unemployment
Average labour productivity
Real wage
Procyclical
Countercyclical
Procyclical
Procyclical
Coincident
No clear pattern
Leading
-
Money supply and inflation
Money supply
Inflation
Procyclical
Procyclical
Leading
Lagging
Financial variables
Stock prices
Procyclical
Leading
Nominal interest rates
Procyclical
Lagging
Real interest rates
Acyclical
Source: Abel, A.B. and Bernanke, B.S. (2001) Macroeconomics, AddisonWesley, p. 288 and Snowdon B. and Vane, H.R. (2005) Modern Macroeconomics, Efward Elgar, p. 306
5. Some regularities in recessions
Another issue is whether output fluctuations have changed their characteristics over
time. About US aggregate fluctuations Romer (2012) expresses the following considerations:
“One can think of the macroeconomic history of the United States since
the late 1800s as consisting of four broad periods: the period before the Great
Depression; the Depression and World War II; the period from the end of
World War II to about the mid-1980s; and the mid-1980s to the present. Although our data for the first period are highly imperfect, it appears that fluctuations before the Depression were only moderately larger than in the period
from World War II to the mid-1980s. Output movements in the era before
the Depression appear slightly larger, and slightly less persistent, than in the
period following World War II; but there was no sharp change in the character
of fluctuations. Since such features of the economy as the sectoral composition of output and role of government were very different in the two eras,
this suggests either that the character of fluctuations is determined by forces
that changed much less over time, or that there was a set of changes to the
economy that had roughly offsetting effects on overall fluctuations” (Romer,
2012, p. 192)
9
Moreover, there are strong regularities in the behaviour of some important macroeconomic variables during recessions. During recessions, Employment falls, unemployment
rises and productivity (as measured as output per worker-hour) declines. The Okun’s law
describes the empirical correlation between shortfalls in GDP and rises in unemployment
rate. In the original formulation a shortfall in GDP of 3% relative to normal growth is
associated with a 1 percentage-point rise in the unemployment rate; more recently the
association 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 an
ongoing basis. NBER researchers determine these dates using a two-step procedure: (i)
local maxima and minima (peaks and troughs) are determined for individual series (with
the help of a computer program, but the ultimate decision is judgemental); (ii) common
turning points across series are identified.
“If, in the judgment of the analysts, the cyclical movements associated
with these turning points are sufficiently persistent and widespread across
sectors, than an aggregate business cycle is identified and its peaks and troughs
are dated. Currently, the NBER Business Cycle Dating Committee uses data
on output, income, employment, and trade, both at the sectoral and aggregate levels, to guide their judgments and dating business cycles as they occur
[NBER (1992)]. These dates are announced with a lag to ensure that the
data 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 the
trend 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 macroeconomic
models (cfr. real business cycle models), in which the source of long-run growth and
short-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 difference
stationary (instead of a trend stationary) process.
Methods to analyse co-movements
• Cointegration
• Common factors
10
1.3
What causes economic fluctuations?
The causes of macroeconomic fluctuations have been hotly debated by the competing
schools of thought. As argued by Romer (2012, pp. 190-191) “the prevailing view is
that the economy is perturbed by disturbances of various types and sizes at more or less
random intervals, and that those disturbances then propagate through the economy. Where
the major macroeconomic schools of thought differ is in their hypotheses concerning these
shocks 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 “rocking
horse” model (Frisch, 1933). The argument is that there are economic behaviours
that, although complex, are intrinsically cyclical.
• There are cycles in the impulse mechanism. Cfr. W.S. Jevons (1835-1882) sunspot
model. One can argue that there are cycles in the agricultural harvest, in technological change, or in the behaviour of policymakers.
• Instead of putting emphasis on regular behaviour, another class of model put emphasis on the randomness of both the impulse and propagation mechanism. As Nelson and Plosser (1982) pointed out, economic output describes a pattern reducible
to 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 propagation
mechanism is characterised as an economic system in which there are no externalities, asymmetric information, missing markets, or other imperfections. Everything
is coordinated by the market. Alternative impulse mechanisms have been hypothesised:
• Unanticipated monetary shocks in Lucas (1975,1977). These are the so-called
“monetary surprise” or “monetary equilibrium business cycle” (MEBC) models.
• 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, unstable demand and investment.
• Technical change (Schumpeterian models).
3. Marshallian models of fluctuations:
• Monetary disturbances (Friedman-Schwarz 1963)
11
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 fully
articulated, artificial economic systems that can serve as laboratories
in which policies that would be prohibitively expensive to experiment
within 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 its
potential usefulness in thinking about reality. Any model that is well
enough articulated to give clear answers to the questions we put to it will
necessarily 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 specific
economic policy as ‘input’ and will generate as ‘output’ statistics describing the operating characteristics of time series we care about, which
are predicted to result from these policies.
2. Representative agent (household/firm) assumption: it is assumed that the economy
is populated by identical individuals. This assumption goes hand in hand with the
neoclassical precept of explaining macroeconomic behaviour as the outcome of individual rationality. As Hoover (1995, p. 38) pointed out “[T]he difficulty with this
approach 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 modelling
of rational behaviour feasible new classical macroeconomics9 adopt “representative agent models, in which one agent or a few types of agents stands in for the
behaviour of all agents.”10
9
The 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”
10
For a criticism, cfr. Hoover (1995, pp. 39-40): “using the representative-agent model ... begs the
question by assuming that aggregation [and interaction] does not fundamentally alter the structure of the
aggregate model. Physics may provide a useful analogy. The laws that relate pressure, temperature, and
volumes of gases are macro-physics. The ‘ideal-gas laws’ can be derived from a micromodel: gas molecules
are assumed to be point masses, subject to conservation of momentum, with a distribution of velocities. An
aggregate assumption is also needed: the probability of the gas molecules moving in any direction is taken
to 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 his
situation. This insight is clearly correct, it is not clear in the least that it is adequately captured in the heroic
12
3. Maximization: households/firms (the representative household/firm) maximize their
utility/profits under constraints. The output produced by firm is determined where
marginal revenues equal marginal costs.
4. Rational expectations hypothesis: agents (the representative agent) form their expectations rationally. As Hoover (1988, pp.14-15) points out, there can be three
interpretations of the REH. A weak form is that “people learn from their mistakes
and do no persist in them.” This raises the question of how people learn, but new
classical macroeconomics does not address this question. A strong form is that
“people actually know the structure of the model that truly describes the world and
use it to form their expectations.” This would imply that economic agents apply
an incredible cognitive power before taking any economic decision. An alternative
interpretation goes back to the original formulation of Muth (1961) and refers to the
relationship between the predictions one can draw from the model and the predictions that the agents within the model can make: the predictions that the modelled
agents are able to do are not worst than the predictions one can imply from the
model (cfr. Muth, 1961). This view leaves open the question of how models relate
to the real world (cfr. Hoover, 1988, p.15). New classical macroeconomics tends to
endorse the second or the third interpretation.
5. Continuous equilibrium: price flexibility ensures continuous market clearing, there
is no moment in which the labour and goods markets are not in equilibrium. In each
market the amount of goods and services demanded is always equal to the amount
produced or made available. As demand and/or supply change, prices change immediately. This assumption is shared with new classical macroeconomic model
Mark I (Lucas Jr, 1972, cfr.). In this setting economic agents (workers, consumers
and 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 is
Pareto-optimal. Unemployment is a voluntary phenomenon (Lucas, 1978). The
assumption of continuous equilibrium (together with the REH) is the most critical and controversial assumption. It is not, of course, shared with Keynesian approaches, but even the monetarist school of Friedman allowed the possibility of
disequilibrium in the short-run. The reduction of macroeconomics to Walrasian
general 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 available production technologies (exogenous shocks to technology). Some RBC models 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 the
behaviour of gases at the macrophysical level, not as derived from the aggregation of molecules of randomly
distributed momenta, but as a single molecule scaled up to observable volume—a thing corresponding to
nothing ever known to nature.” On a similar spirit Kirman (1992) severely criticizes the representativeagent assumption because it fails to fulfil the necessary conditions for perfect aggregation. In this way
the representative agent does not faithful represent the actual individuals, even if their behaviour satisfies
rational choice theory.
13
• consumption smoothing
• lags in the investment process (time to build )
• intertemporal labour substitution
8. Fluctuations in employment mainly voluntary due to the substitutability of work
and 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 theorists
claim that the same theory that explains long-run growth should also explain shortrun fluctuations.
11. Calibration. Calibration is a strategy for finding numerical values for the parameters of the highly idealised models that the RBC school proposes. It consists in
both (i) selecting, through simulation, values for parameters so that the model is
capable of reproducing some statistical properties of the observed economic time
series, and (ii) choosing the parameters of the model from values picked from preexisting microeconomic studies or from general facts about national-income accounting. Calibration eschews standard econometric testing and is also alternative
to the conventional econometric methodology consisting in adequately specifying a
model, finding out the reduced-form, and directly estimating it from the data using
regression techniques. As Hoover (1995, p.29) puts it “[t]he dominance of theory
in the choice of models lies at the heart of the difference between estimators and
calibrators.” With estimation one can compare models from alternative theories to
see which one is more consistent with the data. With calibration “the aim is never
to test and possibly reject the core theory, but to construct models that reproduce
the economy more and more closely within the strict limits of the basic theory....the
real-business-cycle modeller typically does not regard the core theory at risk in
principle. Like the estimators, the calibrators wish to have a close fit between their
quantified model and the actual data —at least in selected dimensions. But the failure to obtain a close fit (statistical rejection) does not provide grounds for rejecting
the fundamental underlying theory (Hoover, 1995, p.29).” In conclusion “[t]he calibration methodology..lacks any discipline as stern as that imposed by econometric
method (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 capital
stock is just the accumulation of this composite commodity. The production function
takes the form:
14
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 labour
and technological progress is said to be labour-augmenting or Harrod-neutral. It is assumed constant return to scale (i.e. the production function is homogeneous of degree
one: F (λK, λAL) = λF (K, AL)) and that the structure of the market is perfectly competitive, so that real wage is equal to the marginal product of labour (mpl) and real interest
rate is equal to the marginal product of capital (mpk).
A Cobb-Douglas production function is usually assumed:
Y = F (K, AL) = K α (AL)1−α ,
0<α<1
(2)
which has the following specificities (that make the model easy to analyse): (i) constant
return to scale; (ii) Y increases with each factor of production ceteris paribus; (iii) the
production function goes to the origin (no free lunch); (iv) diminishing returns to each
factor of production; (v) mpl raises if K increases ceteris paribus, mpk raises if L increases 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 × YL =
L
mpl × K α (AL)
1−α = 1 − α. Analogously, if the real interest rate is equal to mpk then the
K
capital share in output, defined as real interest rate × K
= mpk × K α (AL)
1−α = α. That
Y
is, L−share and K−share are constant. Notice that they remain constant even if the Cobb
Douglas is formulated such that technology is capital augmenting or Hicks-neutral. Indeed there are no difference among these three formulations as far as the Cobb-Douglas
is concerned.
Equation (2) can be written as:
log Y = α log K + (1 − α) log L + (1 − α) log A
(3)
From this equation, taking derivatives with respect of time, one can get:
gy = αgk + (1 − α)gl + z,
(4)
A
where gy , gk , gl are the growth rates of output, capital and labour and z = (1 − α) d log
dt
measures the growth in output that cannot be accounted for by growth in capital and
labour. 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 through
OLS regression. Equation (4) can be rewritten as:
gy + (1 − α)gy − (1 − α)gy = αgk + (1 − α)gl + z
1
α
(gk − gy ) +
z
(gy − gl ) =
1−α
1−α
(5)
(6)
Thus, growth of output per capita depends on the growth of the capital-output ratio
11
mpl =
∂Y
∂L
= (1 − α)K α A1−α L−α ; mpk =
∂Y
∂K
= αK α−1 (AL)1−α
15
and on the Solow residual. The Solow residual has accounted for approximately half the
growth in output in the U.S. between 1870s and 1980s (cfr. Blanchard and Fisher 1989 and
Stadler 1994). This residual has been observed fluctuating significantly over time and has
been described as a random walk with drift plus some serially uncorrelated measurement
errors (Prescott 1986, Stadler 1994). RBC theorists incorporate stochastic fluctuations
in the rate of technical progress into the neoclassical growth model such that this can
display business cycle phenomena. “Thus, RBC theory can be seen as a development of
the 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 = Ktα (At Lt )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 of
depreciation of capital.
• L and K are paid their marginal products. To the m.p.k. it has to be subtracted the
depreciation rate δ. Thus real wages (w) and real interest rate (r) are:
α
Kt
∂Yt
= (1 − α)
At
(9)
wt =
∂Lt
At Lt
∂Yt
rt =
−δ =α
∂Kt
At Lt
Kt
(1−α)
−δ
(10)
• Representative household optimization problem: max expected value of
U=
∞
X
e−ρt u(ct , 1 − lt )
t=0
Nt
H
(11)
where:
– u(·): instantaneous utility function of the representative member of the household;
– ρ: discount rate;
16
– Nt : population; H: number of households;
of the household;
Nt /H: number of members
– c: consumption per member, 1 − l: leisure per member; l: amount each member works;
– c ≡ C/N ; l ≡ L/N .
• Population growth:
log Nt = N + nt
n<ρ
(12)
• Log-linear u(·)
ut = log ct + b log(1 − lt )
b>0
(13)
• Technology path:
et
log At = A + gt + A
(14)
et = ρA A
et−1 + A,t ,
A
−1 < ρA < 1,
where A,t ’s are white-noise disturbances: a series of mean-zero and equal variance
shocks that are uncorrelated with one another.
• Government purchases :
et
log Gt = G + (n + g)t + G
(15)
e t = ρG G
et−1 + G,t ,
G
−1 < ρG < 1,
where G,t ’s are white-noise disturbances.
2.4
Household behaviour
We examine now the consequence that the inclusion of leisure in the utility function and
the introduction of randomness in technology and government purchases have for households’ 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 initial
wealth, and has only one member. In this case U = log c + b log(1 − l). Household’s
budget constraint is c = wl (individual consumption is equal to income and there is no
saving since there is only one period). The Lagrangian for the household’s maximization
problem is
L = log c + b log(1 − l) + λ(wl − c).
(16)
The first order conditions are:
1
1
∂L
= −λ=0 ⇒ λ=
∂c
c
c
17
∂L
b
1
b
1
=−
+ λw = 0 ⇒ (replacing λ with ) −
+ w=0
∂c
1−l
c
1−l c
c
Replacing w with l we get:
1
l=
(17)
b+1
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. Generally
speaking, real wages have two counteracting effects on household’s decision to supply
labour and to consume. The first effect is the income (or wealth) effect: higher real
wages make people feel wealthier, this will tend to suppress the supply of labour (ceteris 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 profitable to replace leisure with work and get an extra income for present or future consumption. 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 substitution effects counterbalance each other, so that the supply of labour is constant. As
regards consumption, there cannot be a substitution effect since the household liven only
one period.
Special case 2: two-periods life
Consider the same case as before except that households live two periods. There is no
uncertainty about the (second-period) interest rate or the second-period wage. In this
case U = log c1 + b log(1 − l1 ) + e−ρ [log c2 + b log(1 − l2 )] and the budget constraint is
c1 + c2 = w1 l1 + r(w1 l1 − c1 ) + w2 l2 (being the return on saving an additional source of
income in the second period). The Lagrangian for the household’s maximization problem
is now
L = log c1 +b log(1−l1 )+e−ρ [log c2 +b log(1−l2 )]+λ[w1 l1 +r(w1 l1 −c1 )+w2 l2 −c1 −c2 ]
(19)
The first order conditions are:
b
b
1
∂L
=−
+ λw1 (1 + r) = 0 ⇒ λ =
∂l1
1 − l1
1 − l1 w1 (1 + r)
∂L
b e−ρ
b e−ρ 1
=−
+ λw2 = 0 ⇒ λ =
∂l2
1 − l2
1 − l2 w2
18
∂L
1
1
=
− λ(1 + r) = 0 ⇒ λ =
∂c1
c1
c1 (1 + r)
∂L
e−ρ
e−ρ
=
−λ=0 ⇒ λ=
∂c2
c2
c2
We get:
1 − l1
1
w2
= −ρ
1 − l2
e (1 + r) w1
(20)
which describes how relative leisure responds to relative wage and real interest rate, and
c1
1
= −ρ
c2
e (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 to
be higher relative to future real wage will increase present labour supply relatively to
future labour supply (increase future leisure relatively to present leisure). A rise in r
increases 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 as
intertemporal substitution in labour supply and have been analysed by Lucas and Rapping
(1969) (cfr. Romer, 2001, p.178).
From equation
(20)itfollows that the elasticity of substitution between leisure in the
two periods
d log
d log
1−l1
1−l2
w2
w1
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 consider to reduce current consumption per member by a certain amount, save that portion of
current 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 expected
benefit equal to the marginal disutility (cost) of decreasing consumption at time t.
Recall that population grows at rate n:
log Nt = N̄ + nt,
where N̄ is a constant. It follows that Nt+1 = en Nt . This can be shown recursively,
starting from period 0:
19
N1 = en N0
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 ) = N̄ + nt,
where log(N0 ) = N̄ .
Marginal utility of consumption (m.u.c.) in period t per member (see again equations
11 and 13 ) is:
∂U
1 Nt
= e−ρt
(22)
∂ct
ct H
Thus the utility cost in decreasing consumption by ∆c is e−ρt c1t NHt ∆c.
The increase in consumption in the next period is equal to ∆c plus the return to saving,
∆c
.
divided by the amount by which population has grown, that is: ∆c+ret+1
n
1 Nt+1
The expected utility benefit is equal to m.u.c. at period t+1 per member e−ρ(t+1) ct+1
H
times the increase in consumption in period t + 1:
−ρ(t+1) 1 Nt+1 ∆c(1 + rt+1 )
Et e
ct+1 H
en
t+1
Since Nt+1 = en Nt and e−ρ(t+1) , NH
, ∆c are not uncertain (the only unknowns are ct+1
and rt+1 ), the latter expression simplifies to:
1
Nt
Et
(1 + rt+1 ) e−ρ(t+1) ∆c
ct+1
H
By equating utility cost e−ρt c1t NHt ∆c to expected benefit we get:
1
1
1
1
−ρ
−ρ
Et
Et [1 + rt+1 ] + Cov
= e Et
(1 + rt+1 ) = e
, 1 + rt+1
.
ct
ct+1
ct+1
ct+1
(23)
This means that there is a trade-off between present and future consumption. How much
is 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 expectation
of the rate of return, and on the interaction of these two variables.
Suppose,
for example,
1
ct+1 is positively correlated to rt+1 . This means that Cov ct+1 , 1 + rt+1 < 0. In this
case, ceteris paribus, ct will be higher (and saving lower) with respect to the case in which
ct+1 and rt+1 are uncorrelated. If c and r are uncorrelated ore negatively correlated (this
latter case would imply correlation between saving and r), then saving today responds
positively 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 consumption. But the more a household works, the less is the leisure it can enjoy. Thus there is a
trade-off between (present) consumption and (present) labour supply. Consider the representative household increasing its labour supply per member by a small amount ∆l to
increase consumption. Again, if the household is behaving optimally the marginal disutility of giving up leisure should be equal to the marginal utility of increasing conusmption.
That is
Nt 1
Nt b
∆l = e−ρt
wt ∆l,
(24)
e−ρt
H 1 − lt
H ct
which reduces to
ct
wt
=
(25)
1 − lt
b
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
1 + rt = α
At Lt
Kt
(26)
1−α
(27)
As Romer (2001, p.181) puts it: “The elimination of government can be justified
on the grounds that doing so allows us to isolate the effects of technology shocks. The
grounds for the assumption of complete depreciation, on the other hand, are only that it
allows us to solve the model analytically.”
Let us call st the fraction of output which is saved, i.e. st = 1 − CYtt . It follows that
ct = (1 − st ) NYtt .
h
i
1+rt+1
1
−ρ
Consider again equation (23): ct = e Et ct+1 . Replacing ct with (1 − st ) NYtt and
taking logs we get:
Yt
(1 + rt+1 )Nt+1
− log (1 − st )
= −ρ + log Et
.
(28)
Nt
(1 − st+1 )Yt+1
21
Since 1 + rt+1 = mpk = α
get:
At+1 Lt+1
Kt+1
1−α
=
αYt+1
Kt+1
and Kt+1 = It = Yt − Ct = st Yt , we
log st − log(1 − st ) = −ρ + n + log α + log Et
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 maximizes
the expected utility, under the two simplifying assumption stated above, the saving rate is
constant.
Consider now equation (25), the other key equation describing household’s optimizing
ct
= wbt . We have:
behaviour: 1−l
t
∗ Yt
log (1 − s )
− log(1 − lt ) = log wt − log b
Nt
(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)Ktα A1−α
L−α
= (1 − α) LYtt
t
t
= (1 − α) ltYNt t .
This yields:
log(1−s∗ )+log Yt −log Nt −log(1−lt ) = log(1−α)+log Yt −log lt −log Nt −log b (32)
After some algebraic manipulation, we get:
lt =
1−α
≡ l∗ .
(1 − α) + b(1 − s∗ )
(33)
Thus optimal labour supply (under the two simplifying assumptions) is also constant. As
Romer (2001: p. 183) puts it: “The reason this occurs despite households’ willingness
to substitute their labour supply intertemporally is that movements in either technology
or capital have offsetting impacts on the relative-wage and interest-rate effects on labour
supply.”
Very schematically:
↑ At =⇒↑
But also
↑ At =⇒↑ s
wt
=⇒↑ lt
Et [wt+1 ]
Yt
=⇒↓ Et [rt+1 ] =⇒↓ lt
Nt
22
Output fluctuations as AR(2)
Recall (we continue to assume G = 0 and δ = 1):
Yt = Ktα (At Lt )1−α
Kt = s∗ Yt−1
et
log At = A + gt + A
log Nt = N + nt
We get:
et + log l∗ + N + nt)
log Yt = α log s∗ + α log Yt−1 + (1 − α)(A + gt + A
Call Yet : deviations of log Yt from the normal path (difference between log Yt and value it
et = 0)
would take if A
et
Yet = αYet−1 + (1 − α)A
(34)
et = ρA A
et−1 + εA,t :
Since A
Yet = (α + ρA )Yet−1 − αρA Yet−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
3.1
Keynesian Theories of Fluctuations
The IS/LM model
The IS/LM model was introduced by John Hicks in an article published in 1937 in Econometrica 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 is
determined by aggregate demand in the short run, i.e. the period in which wages and
prices 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 periodically and not continuously) or menu-costs (i.e. there are costs associated with changing
prices).
The model consists in two parts: the goods market and the money market. The goods
market is described by the IS curve, which denotes a situation in which there is a short-run
equilibrium. IS stands for “Investment-Savings”: in equilibrium planned investment must
be equal to planned savings. The equilibrium in the money market is described by the LM
curve. LM stands for “Liquidity-Money”: in equilibrium money demand, i.e. demand for
liquidity, 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 planned
and 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 , total
taxation 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 and
expected future income, and (1 − ty )Y is disposable income (ty is the portion of
income 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 disposable
income.
• I(r, A) is investment (by firms) as function of real rate of interest r (r influences
negatively 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, which
are ranked by their expected return. If the interest rate falls, then this reduces the
cost of capital and makes some projects profitable that would not otherwise have
been undertaken” (Carlin and Soskice, 2006, p.30). Using a linear investment function:
I = A − ar,
where a is a constant.
Thus we have:
Y D = c0 + cy (1 − ty )Y + A − ar + G.
(37)
1
[c0 + (A − ar) + G]
1 − cy (1 − ty )
(38)
Equilibrium condition:
Y =
Let sy = (1 − cy ) be the marginal propensity to save. We get:
Y =
The term
1
sy +cy ty
1
[c0 + (A − ar) + G]
sy + cy ty
(39)
is called the multiplier. The IS curve is derived graphically in Figure
(4).
12
Actually interest rate is likely to influence consumption of durable goods as well and investment is
likely 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).
(intercept of the IS curve on the Y -axis). For
Notice that for r = 0, Y = cs0y+A+G
+cy ty
c0 +A+G
Y = 0, r =
(intercept of the IS curve on the r-axis). The IS curve as a function
a
of r is indeed
c0 + A + G sy + cy ty
−
Y.
r=
a
a
It is easy to see that (cfr. Carlin and Soskice, 2006, pp. 32 and 63-64):
1. If the multiplier sy +c1 y ty 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 the
multiplier will make the IS flatter: it rotates counter-clockwise from the intercept
on the r-axis.
2. Conversely, a change in a (interest-sensitivity to investment) changes the intercept
on the r-axis, but not the intercept on the Y -axis. Thus, a fall in a rotates the curve
clockwise 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 the
multiplier) 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 equilibrium in the money market for a given price level” (Romer, 2001, p.222). In the following
discussion it is also assumed that the central bank controls directly the money supply.
25
Demand for money:
MD
= L(y, i),
P
∂L
< 0,
∂i
∂L
>0
∂Y
(40)
The demand for money is a decision about the form in which to hold wealth. Wealth
is the set of valuable things, i.e. assets, that you own. While income is a flow, wealth is
a 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 by
writing a cheque, using a debit card or withdrawing money by an ATM machine, you may
have money in a bank account which pays you interest rates or you may own bonds which
also pay interest. Bonds offer you an interest as long as you are prepared to take some
risk.
The allocation of wealth between interest-bearing and not interest-bearing assets will
depend on the interest rate (jointly with your perception of risk and willingness to take
some risk) and the volume of transactions in the economy, which in turns depends on the
level of income. Indeed to carry out transactions you need money. But holding money
bears 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 will
shift the balance in favour of the demand for not interest-bearing
assets. Thus the demand
∂L
>
0
and
negatively
on the nominal interest
for money depends
positively
on
income
∂Y
∂L
rate ∂i < 0 .
Following Carlin and Soskice (2006, p.35), we have
Y
MD
= L(Y, i) = ¯l − li i + T ,
P
v
(41)
where ¯l − li i represents asset demand (i.e. the asset motive for holding money) and vYT
represents transactions demand (i.e. the transactions motive for holding money). The
terms ¯l, li , and v T are here considered constants.
Notice that:
• In the Quantity Theory of Money (well established proposition in classical ecoD
nomics since the XIX century) we just have MP = vYT . v T is the velocity of circulation of transactions, sometimes simply referred to as the “velocity of money”.
• The allocation of wealth between money and interest-bearing assets is also driven
by speculative motives, as underlined by Keynes (1936). Tobin (1958) analysed the
interaction between liquidity preference and behaviour toward risk.
In equilibrium we have that money demand is equal to money supply:
MD
MS
=
P
P
L(Y, i) =
26
MS
P
(42)
(43)
Figure 5: LM curve. Source: Carlin and Soskice (2006, Figure 2.4). Excess supply leads
to a fall in i, excess demand leads to a rise in i.
In the standard IS/LM model it is assumed that M S 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 a
function of Y . We get:
1 1
1 ¯ MS
l−
+
Y
(44)
i=
li
P
li vT
Thus, when Y = 0, i =
1
li
¯l −
MS
P
S
, when i = 0, Y = v T MP − ¯l .
We have that (cfr. Carlin and Soskice, 2006, pp. 38-39,65):
• A rise in v T , the transactions velocity of circulation, moves the intercept of the LM
curve with the Y −axis to the right (while the intercept with the i−axis does not
change): it rotates the LM curve clockwise making it flatter.
• A rise in li , the interest sensitivity of the demand for money, will shift the intercept
with the i−axis towards the origin, while the intercept with the Y −axis does not
change: it rotates the LM curve clockwise making it flatter.
• A change in the money supply will shift the LM curve rightwards by v T (∆M S /P )
(no change in the slope).
• A change in the price level will shift the LM leftwards by v T (M S /∆P ) (no change
in 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
,
E
Pt+1
(45)
E
where r is the real interest rate, i is the nominal interest rate, P is the price level and Pt+1
is the expected price level in the future. Let expected inflation π E be equal to
E
E
have that P π E = Pt+1
− P . Then PP Eπ = 1 − PPE . Hence
t+1
E −P
Pt+1
.
P
We
t+1
P
1
=
E
1 + πE
Pt+1
1+r =
r=
1+i
1 + πE
i − πE
1 + πE
(46)
If π E is low the relationship between i and r is approximated as
i ' r + πE
(47)
Notice that i can be observed, but r (and π E ) 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 effect 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): the
effects of a rise in money supply
29
Figure 7: The liquidity trap. Source: Carlin and Soskice (2006, Figure 2.6).
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