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Lecture Notes BESS 14sept
Macroeconomics and Economic Policy (Università Commerciale Luigi Bocconi)
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Macroeconomics and Economic Policy:
Lecture Notes
Tommaso Monacelli
Bocconi University
tommaso.monacelli@unibocconi.it
https://sites.google.com/site/monacellitommaso/
This version: 14th September 2022
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Modern macroeconomics involves studying supply and demand in multiple markets at a time, rather
than in a single one as microeconomists often do; but importantly where events in each market are allowed
to depend on what is happening, and is expected to happen, in many or even all others. Thus it should
be stressed that there are not different kinds of theories for microeconomics and macroeconomics. Any
differences are fundamentally those of scope of the questions being asked (...).
[K. Athreya, Big Ideas in Macroeconomics, 2013]
I once told a colleague, “If you want to understand geology, study earthquakes. If you want to understand economics, study the biggest calamity to hit the U.S. and world economies (the Great Depression).”
[B. Bernanke]
Then I saw this paper by Lucas [Lucas 1972]. I don’t know why I read it (..) In part, I did because Bob
had been a classmate at Chicago and I had a high opinion of him. I picked up the paper, and it’s talking
about people, two-period lived people. What’s that? There are no people in macroeconomics! I pretty much
saw that the Phillips curve ideas that Sargent and I had been working on were a dead end. That paper
raised the standard, and there was no turning back.
[N. Wallace]
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Contents
1 Introduction
7
2 Keynes and the problem of involuntary unemployment
7
2.0.1
Keynesian macroeconomics . . . . . . . . . . . . . . . . . . . . . . . .
3 Critics to Keynes
16
18
3.1
Friedman and Monetarism . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2
The fundamental limits of policy . . . . . . . . . . . . . . . . . . . . . . . .
22
4 A paradigm shift: from Keynes to Lucas
4.1
26
A methodological revolution . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
4.1.1
Research agenda . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.1.2
Theory and method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.1.3
Equilibrium concept . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
5 Rational expectations and economic policy
31
6 Business Cycles: from disequilibrium to equilibrium analysis
35
6.1
Classification of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
6.2
Imperfect information and rational expectations . . . . . . . . . . . . . . . .
36
6.3
The islands model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
6.3.1
Implications of Lucas aggregate supply curve . . . . . . . . . . . . . .
42
6.3.2
Demand side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
6.3.3
Equilibrium under rational expectations . . . . . . . . . . . . . . . .
44
Lessons from Lucas theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
6.4.1
50
6.4
Limitations and critiques . . . . . . . . . . . . . . . . . . . . . . . . .
7 Modern macroeconomics in brief
7.1
52
Distinction from IS-LM macroeconomics . . . . . . . . . . . . . . . . . . . .
53
8 The intertemporal view
57
9 A two-period model of consumption and saving
57
9.1
Budget constraints of the consumer . . . . . . . . . . . . . . . . . . . . . . .
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9.2
Intertemporal budget constraint . . . . . . . . . . . . . . . . . . . . . . . . .
59
9.3
Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
9.4
Consumer problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
9.5
Experiments with the optimal consumption model . . . . . . . . . . . . . . .
66
9.5.1
Consumption smoothing . . . . . . . . . . . . . . . . . . . . . . . . .
66
9.5.2
Analytical example . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Permanent income hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . .
69
9.6
10 Competitive equilibrium and endogenous real interest rate
70
11 Introducing the government
74
11.1 Ricardian Equivalence theorem . . . . . . . . . . . . . . . . . . . . . . . . .
76
11.1.1 Equivalence between balanced-budget and government debt . . . . .
77
11.1.2 Corollaries of Ricardian Equivalence . . . . . . . . . . . . . . . . . .
80
11.1.3 A tax experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
11.1.4 Effects on private and public saving . . . . . . . . . . . . . . . . . . .
83
11.2 Implications of Ricardian Equivalence . . . . . . . . . . . . . . . . . . . . . .
84
11.2.1 Assumptions under which Ricardian Equivalence holds . . . . . . . .
86
11.2.2 Variations in government spending . . . . . . . . . . . . . . . . . . .
86
12 Production and real business cycles
88
12.1 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
12.2 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
12.3 Equilibrium in the production economy . . . . . . . . . . . . . . . . . . . . .
93
12.3.1 Productivity and investment in general equilibrium . . . . . . . . . .
95
13 The New Keynesian model
99
14 Core with flexible prices
100
14.1 Utility function with leisure . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
14.2 Consumption, leisure and labour . . . . . . . . . . . . . . . . . . . . . . . . . 101
14.3 Nominal spending, money and monetary policy . . . . . . . . . . . . . . . . 101
14.4 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
14.4.1 Marginal cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
14.4.2 Prices and markup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
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14.4.3 Flexible prices and constant marginal cost . . . . . . . . . . . . . . . 105
14.5 Equilibrium employment and output . . . . . . . . . . . . . . . . . . . . . . 106
14.6 The model in compact form . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
14.6.1 Block 1: Labour market equilibrium . . . . . . . . . . . . . . . . . . . 107
14.6.2 Block 2: Firm supply and aggregate demand . . . . . . . . . . . . . . 108
14.6.3 Flexible prices equilibrium and monetary neutrality . . . . . . . . . . 109
14.7 Productivity shock: flexible prices . . . . . . . . . . . . . . . . . . . . . . . . 110
15 Sticky prices, variable markup and monetary policy
112
15.1 Sticky price equilibrium and the role of monetary policy . . . . . . . . . . . 113
15.2 Effects of a productivity shock under sticky prices . . . . . . . . . . . . . . . 114
15.2.1 Optimal response of monetary policy . . . . . . . . . . . . . . . . . . 117
15.3 Effects of a monetary expansion . . . . . . . . . . . . . . . . . . . . . . . . . 120
15.4 The Phillips curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
15.4.1 A more general theory of price setting . . . . . . . . . . . . . . . . . 125
16 Interest rate, expectations and new AD curve
126
16.1 Natural rate of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
16.2 Interest rate, monetary policy and aggregate demand . . . . . . . . . . . . . 129
16.3 Interest rate gap and new IS curve . . . . . . . . . . . . . . . . . . . . . . . 130
17 Rules and discretion in policy-making
137
18 A model for the analysis of monetary policy under inflation targeting
137
18.1 The Phillips curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
18.2 Monetary policy rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
19 Discretionary policy and inflation bias
139
19.1 The problem of the central bank . . . . . . . . . . . . . . . . . . . . . . . . . 141
19.1.1 Graphical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
19.2 Inflation bias and credibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
19.2.1 Commitment: solution to the credibility problem . . . . . . . . . . . 148
20 Stabilization bias and response to shocks
150
20.1 Problem of the central bank in the presence of inflationary shocks . . . . . . 150
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20.2 The PC-MPR system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
20.3 Equilibrium and convergence to the target . . . . . . . . . . . . . . . . . . . 153
20.3.1 Short-term equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 153
20.3.2 Long-term equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 153
20.4 A reduction of the inflation target . . . . . . . . . . . . . . . . . . . . . . . . 154
20.5 A change of inflation expectations . . . . . . . . . . . . . . . . . . . . . . . . 154
20.6 Supply shock and preferences of the central bank . . . . . . . . . . . . . . . 157
20.7 Monetary policy trade-off and algebraic solution . . . . . . . . . . . . . . . . 159
20.8 Expectations and optimal response to the shock . . . . . . . . . . . . . . . . 163
20.8.1 Optimal policy and time inconsistency . . . . . . . . . . . . . . . . . 167
21 Aggregate demand and interest rate rule
168
21.1 New AD curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
21.2 Optimal interest rate rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
21.2.1 Optimal rule vs. Taylor Rule . . . . . . . . . . . . . . . . . . . . . . 169
22 The return of liquidity traps
173
23 A model of the zero lower bound
174
23.1 Paradoxes at the zero lower bound . . . . . . . . . . . . . . . . . . . . . . . 182
.
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1
Introduction
These notes are a (necessarily incomplete) introduction to modern macroeconomics. In order
to understand what “modern macroeconomics” really is, it is of paramount importance to
first take a historical perspective. Macroeconomics, in fact, is a subject that proceeds for
cumulative knowledge, with recurrent revolutions and changes of paradigm.
The history of macroeconomics can be divided into two main eras (De Vroey 2018). The
first one, roughly extending from 1940s to the 1970s, comprises the seminal contribution of
Keynes (“The General Theory”) and the so-called “Keynesian macroeconomics” (of which
the well-known IS-LM model is a typical example). The second era started between the end
of the 1960s and the mid-seventies with the revolution initiated by E. Phelps, M. Friedman
and B. Lucas.
Our historical perspective will develop through the lens of three scientific paradigms
(and associated revolutions):
1. Keynes and his view of the causes of unemployment.
2. The natural rate hypothesis of Phelps and Friedman.
3. The Lucas neoclassical revolution.
2
Keynes and the problem of involuntary unemployment
With his General Theory (1936), J.M. Keynes is considered the founding father of macroeconomics as a field. Keynes’ aim in writing his book was to identify the causes of the mass
unemployment that affected all the developed economies during the Great Depression that
started in 1929 in the US with a major banking crisis and held sway in several countries
until the late 30s. Macroeconomics arose in the wake of the Great Depression from the
desire to bring to the fore the existence of market failures on which the state should act.
Unemployment was considered the main of these market failures.
The Great Depression was a severe worldwide economic depression that took place
mostly during the 1930s. The timing of the Great Depression varied across nations; in most
countries, it started in 1929 and lasted until the late 1930s. It was the longest, deepest,
and most widespread depression of the 20th century. The Great Depression started in the
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United States after a major fall in stock prices that began around September 4, 1929, and
became worldwide news with the stock market crash of October 29, 1929 (known as Black
Tuesday). Between 1929 and 1932, worldwide gross domestic product (GDP) fell by an
estimated 15%. By comparison, worldwide GDP fell by less than 1% from 2008 to 2009
during the Great Recession. Some economies started to recover by the mid-1930s. However,
in many countries, the negative effects of the Great Depression lasted until the beginning
of World War II. In the United States, where the crisis began, real GDP fell by about 25
percent between 1929 and 1933 (Figure 1).
Figure 2 illustrates the magnitude of the increase in unemployment between 1929 and
1933 in the United States, compared to the one during the recent so-called Great Recession
of 2007-08. Figure 2 displays also the behavior of unemployment during the so-called Long
Depression of the 1890s. The Long Depression was a worldwide price and economic recession,
beginning in 1873 and running either through the spring of 1879, or 1896, depending on the
metrics used. It was the most severe in Europe and the United States, which had been
experiencing strong economic growth fueled by the Second Industrial Revolution in the
decade following the American Civil War. The episode was labeled the “Great Depression”
at the time, and it held that designation until the Great Depression of the 1930s. In all three
cases the impact on unemployment was large and persistent over time.
The Austrian view. When the Great Depression started the main diagnosis about
the crisis available to economists was coming from the Austrian school. The Austrian view
held that the crisis was the result of overinvestment and misallocation of resources. This
state of affair required, for its solution, a process of “liquidation,” which could only take place
through a substantial and generalized deflationary process, i.e., a downward adjustment of
prices and wages.
Keynes and the role of government. However, when the depression persisted for so
many years (with a large increase in unemployment) and despite a strong downward pressure
on wages, economists started to criticize the virtues of “laissez faire” and became vocal
about the importance of government intervention. The work of Keynes was enthusiastically
received in the early thirties precisely because it tried to rationalize, in an alleged “general
theory,” the root causes of unemployment, providing a more rigorous foundation to the
widely held view that more government intervention was needed to address to the crisis.
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Figure 1: Real GDP during the Great Depression in the United States (1929 = 100)
Figure 2: Unemployment rate in the United States during the Great Depression vs. the
Great Recession of 2007-08 vs the Long Depression of 1890s (source Duca 2017).
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Unemployment and wages in Keynes theory. A detailed account of the General
Theory is well beyond the scope of these notes. It would in any case be a difficult enterprise.
According to many economists, and despite its historical impact, the work of Keynes remains
difficult to decipher, full of ambiguities, not least because Keynes refrained completely from
using formal models and mathematics to clarify his main concepts. We will therefore limit
our attention to one specific, albeit crucial, dimension of Keynes’ thought, i.e., his view of
the causes of involuntary unemployment and its relationship to wage flexibility.
Keynes’ fundamental goal was to understand why (massive) unemployment could occur
despite the presence of widespread willingness by workers to supply labor. From here the
label “involuntary.” Keynes was skeptical of the “classical view,” also held be the Austrian
theory, whereby the cause of unemployment is the downward inflexibility of wages. The
experience of the Great Depression, in the US, the UK and several other countries, was in
his view a blatant falsification of the classical theory. Keynes held that mass unemployment
was somehow related to a “failure of market capitalism.” To better understand his view, it
is useful to contrast the “classical” and the “keynesian” view of wages, employment, and of
the functioning of the labor market.1
Labor market and employment: the classical view. Formally the classical view of
the labor market comprises two elements: (i) a labor demand (LD) schedule, and (ii) a labor
supply (LS) schedule. Both relationships are depicted in Figure 3.
The LD schedule describes a downward sloping relationship between the real wage and
employment. In principle it could be derived by the profit maximizing choice of a perfectly
competitive firm. Assume that firm’s output is produced via the production function
′
′′
y = F (n), with F > 0 and F < 0
(1)
In order to maximize profits, a perfectly competitive firm equates the real wage wr to the
marginal product of labor:
′
wr = F (n) ≡ mpn
(2)
where wr is the real wage (in logs), and mpn denotes the marginal product of labor. Notice
′′
that, given F < 0, the marginal product of labor is an inverse function of employment n.
1
See Gali (2016).
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Figure 3: Labor market equilibrium in the classical view
The LS schedule can be derived from the optimal labor supply choice of a representative
household equating the real wage to the marginal rate of substitution (mrs) between leisure
and consumption. We will elaborate more on this in a later section. For now it suffices to
assume that the worker, at the optimum, and for any given level of consumption, will supply
more labor at a higher real wage.
At a given real wage w0r , labor demand and supply can in principle diverge. But in the
classical view, the corresponding (so-called “Walrasian”) equilibrium entails that the wage
∗
adjusts to the level wr so that both equations—labor demand and labor supply—are satisfied simultaneously. Therefore, in the classical view, and in the absence of other restrictions
to wage adjustment, there is no possibility of involuntary unemployment. The real wage will
adjust in order to insure that labor demand and supply will always be equated. Unemployment will emerge in a classical environment only if, due to the effects of collective bargaining
or other legal or institutional constraints, the prevailing wage lies above its Walrasian level.
Employment is then determined by labor demand, and falls short of the quantity of labor
supplied at that wage. In that case, a fraction of individuals will be jobless despite their
desire to work—that is, involuntary unemployment would emerge.
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In short, the classical view of the labor market can be summarized as follows:
• Labor demand and labor supply are determined by the real wage.
• Wage flexibility will insure that labor demand and supply will coincide at the equilibrium.
• At a given real wage above the equilibrium (walrasian) level, involuntary unemployment
can emerge.
Labor market and employment: the keynesian view. Keynes held a fundamental
objection to the classical theory of employment. He criticized the assumption that employment is determined by the real wage, without regard to aggregate demand conditions in the
economy. In the Keynesian theory of employment the real wage is determined by employment, not the other way around. In short, according to Keynes, employment is determined
by the quantity of output that firms want to produce. In turn, desired output is a function of
the aggregate demand for consumption or investment in the economy. Hence labor demand
is independent of the real wage.
Formally, the keynesian view of the labor market comprises three building blocks:
1. labor demand schedule (LD)
2. wage schedule (WS)
3. labor supply schedule (LS)
The LD schedule is represented in Figure 4. Given that labor demand is independent
of the real wage, the LD schedule is vertical.
The second building block is the wage schedule (WS). In Keynes’ view firms have some
monopolistic power in setting their price. Their pricing behavior can be represented (in
logs):
p = µ + (w − mpn)
| {z }
(3)
nominal
marginal cost
Firms set their price as a markup µ over their nominal marginal cost of production,
given by the difference between the nominal wage w and the marginal product of labor mpn.
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Figure 4: Wage schedule in Keynes view. A given level of labor demand nd0 determines the
real wage.
Notice that under perfectly competitive markets µ = 0, and (3) reduces to (2). Rearranging
we can write:
w − p = −µ + mpn
(4)
Since the marginal product of labor mpn is a decreasing function of employment (from (2)),
we can represent the wage schedule WS as a downward sloping curve. In Keynes’ logic,
exogenous conditions in aggregate demand (for consumption, private and public investment,
government spending) determine the level of labor demand by firms (indicated by nd in the
figure). Thus, at a given level of labor demand nd0 , and given the markup, equation (4)
determines the real wage.
The critical implication of Keynes’ view is that involuntary unemployment can emerge
in equilibrium, as illustrated in Figure 5. A given level of labor demand nd0 (in turn determined by the level of aggregate demand in the economy) determines the real w0r . In turn,
at that real wage w0r , labor supply (i.e., workers’ willingness to work) exceeds the demand
of labor, thereby generating involuntary unemployment. In Keynes’ view there is no auto-
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Figure 5: Involuntary unemployment in the keynesian view. A given level of labor demand
nd0 determines the real wage wr0 . At wr0 the supply of labor exceeds the demand of labor.
matic mechanism (e.g., via a real wage adjustment) that can guarantee the labor market to
return to full employment. If labor demand is insufficient - because of depressed aggregate
demand conditions - then the real wage remains too high to guarantee full employment (i.e.,
equalization of demand and supply of labor).
Unemployment and government intervention. It follows that, in Keynes’ view,
the critical determinant of unemployment is the lack of aggregate demand. Therefore the
only policy intervention that can address the problem of unemployment is an expansion of
aggregate demand, typically via an expansion of government purchases and public investment,
or via an increase in the supply of money. This in turn will cause an expansion in firms’ labor
demand, as illustrated in Figure 6. This is probably the most well-known Keynes’ message
in economics, and also the one that had the largest impact on the political and economic
debate in the subsequent decades.
Given his aggregate demand theory of unemployment, Keynes was critical of any policy intervention aimed at addressing the unemployment problem via an increase in wage
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Figure 6: Addressing unemployment in the keynesian view: an expansion in aggregate demand causes a rise in labor demand.
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flexibility. According to Keynes, the favorable effect on employment of any labor market
intervention aimed at reducing the real wage would depend on whether such a reduction increases firms’ investment, i.e., a component of aggregate demand. In turn this would depend
on whether a lower real wage would increase the marginal product of capital or decrease the
real interest rate. According to Keynes there is no reason whatsoever for a lower real wage
to stimulate the marginal efficiency of capital, and therefore stimulate investment. Keynes’
conclusion is therefore that too high wages are not the root cause of unemployment. To
the contrary, and especially in the context of widespread involuntary unemployment (such
as during the Great Depression), “wage rigidity is a good thing,” because it in principle
safeguards the purchasing power of workers.
2.0.1
Keynesian macroeconomics
Keynes’ General Theory (GT), and his theory of unemployment, had an extraordinary impact. It changed the course of economic theory “by setting the scene for a new discipline,
macroeconomics, that is simplified, applied, and policy-oriented general equilibrium.”2 In
the context of the Great Depression many (British) economists had come to think that wage
cuts had to be opposed and public works were needed to prop up employment. Keynes’
theory provided a rationale for this viewpoint.
For all the enthusiastic reception of the GT, in the wake of its publication the confusion
over its central message was however great. A famous session of the Econometric Society
Conference in 1937 was devoted to the GT, and three important papers discussing it were
presented, respectively by J. Meade, R. Harrod and J. Hicks. Among these contributions,
the one by Hicks was to have the most extraordinary impact, because it laid the foundations
of the famous IS-LM model. The IS-LM model marked a real split between “the economics
of Keynes” and so-called “Keynesian economics.”3 The same conference, however, and especially on the basis of Hicks’ paper, marked the beginning of a (mis)interpretation of Keynes’
theory of unemployment. “Keynesian economists” came to declare that the hallmark of
Keynesian macroeconomics was the wage rigidity assumption. The current version of the
IS-LM model, in fact, is centered on the fundamental, and logically similar, assumption of
(complete) nominal price rigidity. However in Keynes’ GT the wage rigidity assumption is
never explicitly mentioned and in many instances even declared not to be a necessary con2
3
De Vroey (2018).
See J. Hicks “Mr Keynes and the Classics” (1937).
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dition for the emergence of unemployment. The debate on whether Keynes really believed
in wage rigidity is still undergoing.
The opaque interpretation of Keynes’ view on wage rigidity persisted. It even led Hicks
to interpret Keynes’ GT simply as a special case of the classical theory, emerging precisely
in the special circumstance in which wages are assumed to be rigid as opposed to flexible.
According to Hicks, the real novel contribution of Keynes’ GT, relative to the classics, lied
in its theory of liquidity traps, i.e., depressed states of the economy in which the standard
remedy against unemployment of expanding the money supply does not work, so that the
only possible solution is to rely on expansionary fiscal policy.4
Keynesianism and the Phillips curve. In the 1950s and 1960s most macroeconomists
were Keynesian. One unifying pillar of so-called Keynesian economists was the principle that
(some version of) the Phillips curve (PC) had to be incorporated into the main macroeconomic apparatus. The PC, originally put forward by the New Zealand economist William
Phillips, was the empirical observation of a (long-term) negative relationship between the
rate of nominal wage variation and rate of unemployment. This observation, grounded in the
data, led many economists of the time to view the PC as a stable relationship. Intuitively,
the existence of a PC would suggest that whenever the labor market is tight unemployment
is low and wages tend to rise, the reverse being true in the case of a slack labor market displaying unemployment. This view led some economists to take the further step of declaring
that any positive increase in the rate of growth in wages would be conducive to inflation,
thereby delivering the idea that the PC represented a stable relationship between (price)
inflation and unemployment. It is important to clarify that the “PC idea” was not originally
formulated by Keynes, but became part of the “Keynesian consensus” during the 1950s and
the 1960s. This prompted a further step taken by (nobel prize winners) P. Samuelson and
R. Solow who declared that the PC “offered a menu for policy-making.” In their view, once
armed with the PC, economists can provide the government with a menu of options: a decrease in unemployment is achievable only at the cost of a positive (and increasing) inflation
rate.
Formally the “keynesian” version of the Phillips curve would state:
π = −α u
4
See later for our treatment of liquidity traps.
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where π is the rate of price inflation, u is the unemployment rate, and α > 0 is a parameter.
The PC rapidly became an integral element of the Keynesian apparatus, for two main
reasons. First, it provided economists with the possibility of making clear policy recommendations to the government about the different mixes of inflation and unemployment.
Second, it filled a gap in the traditional keynesian theory by providing a theory of how
goods prices evolve over time. In the words of E. Phelps (2006): “This seemingly rock-solid
Phillips curve put the American Keynesians into a sort of euphoria, as if they had discovered
atomic power.” The PC, however, was at the time hardly more than an empirical observation. Put differently, its logic lacked foundations in microeconomic theory. A fundamental
change in the intellectual scene, a true scientific revolution due to E. Phelps and M. Friedman, happened in the late 1960s precisely when economists tried to provide the PC with
microeconomic foundations. This happened with the introduction of the so-called “natural
rate hypothesis.” We will return to this issue below.
3
Critics to Keynes
3.1
Friedman and Monetarism
During the 1960s the Keynesian consensus came to be the object of a fierce attack, the
so called monetarist counter-revolution. The key contribution of Monetarism was the 1963
hallmark book by Milton Friedman and Anna Schwarz (F-S) “A Monetary History of the
United States.” Its most famous part was chapter 7 entitled “The Great Contraction of
1929-33.” F-S famously argued that the Keynesian claim whereby the Great Depression was
the manifestation of a large-scale failure of the market system due to insufficient demand
and investment had to be dismissed. In their view the responsability of the depression lied
in the Federal Reserve’s restrictive monetary policy in the 1930s, which failed to provide
adequate liquidity to the banking system, thereby precipitating rather than easing an ongoing
recession.
As hinted above, the Great Depression was viewed by Keynes as a prototypical example
of a liquidity trap, in which monetary policy becomes impotent in stimulating economic
activity and fiscal policy becomes the only toolkit for economic policy. F-S had a different
view. In their Friedman’s words: “The Fed failed to exercise its responsabilities assigned
to it by the Federal Reserve Act to provide liquidity to the banking system. The Great
Contraction is tragic testimony to the power of monetary policy - not, as Keynes and so
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many of his contemporaries believed, evidence of impotence.” (Friedman 1968)
Figure 7 and 8 illustrate Friedman’s point. The figures show the behavior of the monetary aggregate M2 during the Great Depression vs the one during the 2007-08 Great Recession. M2 fell by one-third in the first 5 years of the Great Depression, but rose throughout
the Great Recession. Similarly the U.S. monetary base (a more direct index of the monetary policy activism by the Fed) grew much more in the Great Recession than in the Great
Depression.
Based on the legacy of F-S’s work, Bernanke famously stated in a 2002 (i.e., prior to
the Great Recession) speech in honor of Milton Friedman: “Let me end my talk by abusing
slightly my status as an official representative of the Federal Reserve. I would like to say
to Milton and Anna: Regarding the Great Depression. You’re right, we did it. We’re very
sorry. But thanks to you, we won’t do it again.” (Bernanke 2002).
In 2005 Bernanke was then appointed chairman of the Fed, therefore with the main
responsability of the conduct of monetary policy. Figure 7 shows that he actually avoided
repeating the same (alleged) mistakes of the Fed during the Great Depression. This is one
of the reasons, according to many, for why the increase in unemployment during the Great
Recession turned out to be much smaller than under the Great Depression, as previously
illustrated in Figure 2.
In short, Friedman’s point about the Great Depression was that it wasn’t an inevitable
result of the Great Crash of 1929. Rather, it was an avoidable consequence of the Fed’s
reaction to it. That Fed reaction being to let all the banks go bust and allow the money
supply to shrink. This lesson was vividly clear to Bernanke when he held the responsability
of chairing the Fed during the financial crisis of 2007-08. The behavior of the Fed during
that period, in fact, was radically different, precisely thanks to the policy lessons learnt by
Bernanke from the F-S work. Despite that, the consequences during the Great Recession
have been dire for the US and the world economy, both in terms of GDP contraction and
rise in unemployment (Figure 2). This has led Bernanke to state, in 2014, that “September
and October of 2008 was the worst financial crisis in global history, including the Great
Depression.”5
5
“September and October of 2008 was the worst financial crisis in global history, including the Great
Depression,” Mr. Bernanke is quoted as saying in the document filed with the US Court for American
claims. Of the 13 “most important financial institutions in the United States, 12 were at risk of failure
within a period of a week or two.” Asked why he thought it was essential for the government to rescue
AIG (the financial insurance company), Bernanke said, “AIG’s demise would be a catastrophe” and “could
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Figure 7: M2 in the United States: Great Depression vs. Great Recession
Figure 8: Monetary base in the United States.
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Figure 9: Federal budget surplus in the US: Great Depression vs. Great Recession.
This is however not the full story. Figure 9 displays the evolution of the Federal budget
surplus during the Great Depression (starring in 1929) vs the Great Recession (with 2006 as
a starting date). Clearly also the behavior of fiscal policy has been different during the last
crisis relative to the Great Depression, with fiscal policy running a much larger deficit from
2006 onward, therefore exerting a much more expansionary effect on the economy relative
to the Great Depression period.
Financial factors during the Great Depression. The debate surrounding the causes
of the Great Depression evolved after Friedman and Schwarz. A new complementary view,
by Mishkin (1975) and Bernanke (1983), was developed, which put at center stage the role of
financial factors in making the depression simultaneously so deep and prolonged. Bernanke
argued that the epicenter of the crisis was the collapse in the financial system due to a run
on intermediaries. As more recently argued by the same Bernanke, the financial view of the
have resulted in a 1930s-style global financial and economic meltdown, with catastrophic implications for
production, income, and jobs.”
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Great Depression can be described in terms of two, not mutually exclusive, and potentially
overlapping, channels through which a financial crisis can depress economic activity:
1. Household balance-sheet view. A buildup in (household) debt, in combination with the
collapse in asset prices, especially housing prices, leading to a contraction in households’
net worth and in turn in spending. This view is often related to the so-called “debtdeflation” hypothesis. This hypothesis argues that recessions and depressions are due
to the overall level of debt rising in real value because of deflation, causing people to
default on their consumer loans and mortgages. The theory was developed by Irving
Fisher following the Wall Street crash of 1929.
2. Fragilities in the financial system. This view emphasizes the role of financial intermediaries. Excessive risk-taking and reliance on short-term funding by banks (either
via traditional short-term deposits or wholesale funding in the interbank market) can
result in panic, (bank) runs and credit crunches.
As hinted above, the two views are not mutually exclusive. The first view focuses the
attention on the role of household debt, housing prices and its implications for consumer
spending. The second view focuses on the impaired balance sheet of banks which may cause
a panic by investors and depositors, leading to a collapse in the ability of the financial
system to provide credit to the economy. The “financial view” of the Great Depression
(comprising both 1 and 2 above) largely informed Bernanke during the recent crisis. The
panic of 2008 differed from the Great Depression of the 1930s in that the runs on the financial
system during the recent episode were on wholesale funding (i.e., the interbank market where
intermediaries borrow and lend funds to each other), and occurred electronically, while in
the 1930s retail depositors lined up in the streets. But the overall effect was the same: a
loss of confidence in credit providers caused the supply of credit to plummet, the external
finance premium (i.e., the interest premium required by lenders to provide funds to “risky”
borrowers as opposed to “safe” borrowers, such as the government) to spike, and the real
economy to contract rapidly.
3.2
The fundamental limits of policy
Until 1959, the Keynesian doctrine suggested that it was logically possible to reduce unemployment to virtually zero: the recipe was to expand aggregate demand (i.e., how much
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agents want to spend and/or invest) permanently, either via expansions of government purchases or via expansions of the central bank’s money supply. As hinted above, the “keynesian” Phillips curve would give policymakers a menu of choices between the cost of inflation
and the benefit of lower unemployment.
Still today, a popular view among the public, emerging especially in the aftermath of
severe recessions, is that fiscal and/or monetary policy could be used to continuously push
the level of economic activity upward (and the unemployment rate downward). This would
spare the economy the loss of jobs and the inefficient utilization of capital that typically characterize downturns. The mere acknowledgment of the role of agents’ expectations, though,
poses a tight limit to this view. We will first explain this fundamental point through the aid
of an example.
Expectations and monetary neutrality. Consider the example of an exogenous
increase in the quantity of money (conducted by the central bank) by 5 percent, with the
aim of boosting real economic activity (or reduce unemployment). To highlight the role of
expectations as clearly as possible, and their implications for the conduct of economic policy,
it is crucial to distinguish between two polar cases: expected and unexpected increases in the
money supply.
Let’s begin with the case of an expected increase in the quantity of money. Since firms
perfectly anticipate an increase in the amount of money of 5 percent, what is optimal for
them is indeed to increase prices by 5 percent, and not production: for a given value of
nominal wages, in fact, their profits would increase without further costs of production.
Theoretically, in the limit case in which firms correctly anticipate an increase in the quantity
of money of 5 percent each year, all they have to do is to steadily increase prices each year
by 5 percent. What would households do? With fixed nominal wages, and rising prices, the
purchasing power of wages would be curtailed (i.e., the real wage would fall), and therefore
households would not be willing to supply the same amount of labor. In order to maintain, in
equilibrium, the same level of employment and production, it would be necessary to increase
also nominal wages by 5 percent. But if all firms behave in the same way, it becomes
impossible for monetary policy to affect economic activity. A perfectly anticipated increase
in the growth rate of money of 5 percent each year would translate only in a 5 percent price
and wage inflation, with no effects whatsoever on real variables such as output, consumption
and employment. The last proposition is what is known as monetary neutrality.
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In reality, however, firms often do not perceive exactly, or are caught by surprise by,
the increase in the quantity of money. So at least part of the monetary expansion is translated into higher production. The logical implication of this statement is therefore that,
for monetary policy to be able to affect the level of economic activity via an expansion of
the quantity of money, any monetary expansion should be unexpected, i.e., it should catch
economic agents by surprise. Continuing with our previous example, the central bank should
therefore begin to surprise firms with a growth rate in the money supply in the next year of,
say, 7 percent.
The key question is: could the monetary policy authority conduct such a “surprise
policy” systematically? In other words, could monetary policy try to surprise agents with
systematically higher rates of money growth in order to continuously increase the level of
output? It is not hard to appreciate that, sooner or later, firms would incorporate in their
expectations this new rate of money growth and therefore in actual prices. Because of this
shift in expectations, continuous further increases in money growth would end up translating
only into higher inflation, with no effects on economic activity.
The main conclusion of the analysis above is therefore that the ability of monetary
policy to affect economic activity is heavily constrained by agents’ expectations about the
same policy’s course of action. The role of expectations in shaping the limit of economic
policy was first discussed in the famous American Economic Association (AEA) inaugural
address by Milton Friedman in 1968. With some caveats, it can be considered as the date
of birth of modern macroeconomics. The ideas of Friedman (1968) were simultaneously also
developed by E. Phelps and then formalized and systematized by the true founders of modern
macroeconomics, the so called “new neoclassicals” B. Lucas, T. Sargent, and E. Prescott.
The natural rate hypothesis. In his 1968 American Economic Association address,
Friedman precisely discusses the limits (or lack thereof) of monetary policy in affecting the
level of economic activity, in particular the rate of unemployment.6 The (somewhat revolutionary) argument of Friedman was based on two main pillars. The first pillar, as already
discussed above, is the role of expectations. The second pillar derives from acknowledging
the role of the “long-run.” Friedman suggests that in the long run the rate of unemployment does not deviate from a natural rate. Previously Samuelson and Solow (1960) viewed
the long run as merely the consequence of a series of Keynesian short runs. In contrast,
6
For an extensive analysis of the relevance of Friedman’s presidential address see Mankiw and Reis (2017).
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Friedman viewed the long run as the time frame under which we should apply the principles
of classical economics, especially monetary neutrality. Some more recent interpretations of
Friedman’s dictum view the natural rate of unemployment as the frictional one determined
by the structure of the labor market (institutions, degree of competition, unions’ power),
and however still independent of the effect of monetary policy.
The mere formalization of the two pillars of Friedman’s address (the role of expectations in affecting policy and the long run view) yields a relationship between inflation and
unemployment that can be described as follows:
π = π e − α(u − un )
(6)
where π e is the expected rate of inflation and un is the natural rate of unemployment.
Notice that the version of the PC stated in (6), which can be labeled Friedmanian PC,
is radically different from the keynesian PC of (5). Equation (6) states that the rate of
unemployment deviates from its natural level only to the extent that inflation deviates from
expected inflation. In other words, it is only surprise inflation (the difference π − π e ) that
is associated to deviations of unemployment from its natural level. This is precisely in line
with our discussion above on the role of expectations and the limits it imposes on the ability
of monetary policy to affect economic activity.
An equivalent interpretation of (6) is that, at given inflation expectations, there exists
a tradeoff between inflation and unemployment. The main insight of Friedman and Phelps
is that, if the monetary authority aims at lowering unemployment below the natural rate by
expanding the money supply, and therefore inflation, the tradeoff can be “exploited” only
by catching agents by surprise, but not systematically. The reason is that, if policymakers
were to try to exploit the tradeoff systematically, inflation expectations would adapt, and the
Phillips curve relationship would shift. Equation (6), in fact, suggests that a Phillips curve
exists only in the short-run. In the long run, or equivalently said “on average,” inflation
is equal to expected inflation, unemployment is equal to its natural rate, and there is no
significant relationship between inflation and economic activity. In a nutshell, these are the
dictums of (long-run) classical economics.
How are expectations formed? Importantly, in equation (6), nothing is specified
about how expectations are actually formed. For instance, Friedman (1968) did not discuss
a theory of expectations, but only qualitatively referred to “slowly changing expectations.”
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If expectations are slow to adjust, then unemployment can temporarily deviate from the
natural rate, generating a short-run tradeoff between inflation and unemployment. In the
long run, however, since inflation expectations cannot systematically deviate from the actual
level of inflation (as per our discussion above, the monetary authority cannot systematically
surprise agents) π = π e will hold, and therefore u = un , so that there is no permanent tradeoff
between inflation and unemployment. The specification of a coherent model of expectations
formation, put forward by Robert Lucas, contained the seeds of the subsequent intellectual
revolution.
Relevance of Friedman’s paper. As noted by Gordon (2009), the timing of Friedman’s paper was exactly right. For two main historical reasons. First, the fiscal expansion
undertaken by Kennedy and Johnson, with both tax cuts and more spending for the Vietnam
war, accompanied by an expansionary monetary policy by the Federal Reserve, had pushed
the US unemployment rate from 5.5 percent down to 3.5 percent.
As shown in Figure 10, however, each year between 1963 and 1969 the inflation rate had
accelerated, precisely as predicted by the Friedman’s view. Second, the Phelps-Friedman’s
theory predicted that if governments kept printing money in order to decrease unemployment, this would just result in an increase in inflation. The surfacing of stagflation in the
1970s - a combination of higher inflation and higher unemployment (lower output growth)
resulting from higher oil prices - was perceived as an empirical confirmation of the natural
rate hypothesis.
4
A paradigm shift: from Keynes to Lucas
After the Second World War business cycle research was dominated by Keynes’ followers.
However, the natural rate hypothesis formulated by Friedman and Phelps, and the simultaneous occurrence of the stagflation episode during the 1970s, marked a profound crisis in the
pillars of keynesian macroeconomics. In addition to these factors, a third methodological
reason contributed to the scientific decline of keynesian macroeconomics. In the keynesian
view, business cycles (expansions and contractions in economic activity) were seen as disequilibrium phenomena. Disequilibrium has a twofold meaning. For one, business cycles were
viewed as the result of “mistakes,” especially by monetary and fiscal policymakers; a type
of pathology that could well be corrected if only economic policy were conducted soundly.
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Figure 10: Inflation and unemployment during the 1960s and 1970s
Furthermore, and more formally, disequilibrium referred to the assumption that important
variables in the analysis, for instance prices and wages, are exogenously fixed and not explained within the model. This meant that the supply of labor in the labor market and
the supply of goods in the goods market are rationed. What does rationing mean? In the
traditional Keynesian macroeconomics (although, as explained earlier not necessarily in the
original Keynes’ work) the wage is given (fixed, or exogenous). At that given wage, firms
demand some amount of labor (say 100 units). If workers willing to work are more than 100,
the supply of labor is rationed, i.e., 20 workers do not find a job. A similar logic could be
applied to the goods market, and to the supply and demand of goods. In some cases, in the
Keynesian tradition, prices and wages were assumed to be mechanically adjusted to the level
of excess supply in each market, such that price and wage inflation was a decreasing function
of the rate of unemployment. This was the “Keynesian way” to generate the Phillips curve.
In the late 1960s and early 70s, however, the Keynesian approach started to be heavily criticized for mechanically postulating such relations, without giving them rigorous theoretical
explanations. In the jargon of modern macroeconomics, such keynesian relationships did not
feature any type of microfoundation.7
7
This criticism is best summarized in a famous paper by Lucas and Sargent (1978).
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To summarize, three main factors contributed to the decline of keynesian macroeconomics in the 1970s:
1. Natural rate hypothesis.
2. Stagflation in the main industrialized countries.
3. Lack of rigorous theoretical foundations.
In the early 1970s Robert Lucas launched the rational expectations revolution with a series of fundamental papers. As argued by Mishkin (1995): “Ever since then macroeconomics
has never been the same.”8
Relative to Keynesian macroeconomics, Lucas’ work had a substantive and methodological component. For one, it focused on explaining business cycle fluctuations in general,
not necessarily focusing on special episodes like the Great Depression. In this vein, his purpose was to allow more general and better-founded discussions of economic policy. From
a methodological viewpoint, Lucas’ contribution consisted in laying out a new scientific
paradigm, whose pillars consisted in the following: dynamic analysis, general equilibrium,
microfoundations, and rational expectations. We will analyze these features below.
4.1
A methodological revolution
Lucas’ work marked a turning point in the history of macroeconomics, a true methodological
revolution. The revolution consisted of major innovations in three main areas:9
1. Research agenda.
2. Relationship between theory and method.
3. Concept of equilibrium.
8
Lucas won the Nobel Prize in Economic Sciences in 1995 “for having developed and applied the hypothesis of rational expectations, and thereby having transformed macroeconomic analysis and deepened our
understanding of economic policy” (Royal Swedish Academy of Sciences, 1995).
9
De Vroey (2018).
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4.1.1
Research agenda
As already mentioned above, macroeconomics arose in the wake of the Great Depression from
the desire to emphasize the existence of market failures on which the state should intervene.
With the paradigm change introduced by Lucas, business cycle fluctuations became the
central object of macroeconomics. The involuntary unemployment problem caused by major
crisis like the Depression was not considered anymore a priority. Research was redirected
towards studying the determinants of the level of economic activity, i.e., the total number of
hours worked. This change of view was the result partly of the perceived shortcomings of the
keynesian agenda and partly of the fading of the memories of the Great Depression. Lucas
shifted the attention towards explaining business fluctuations, arguing that they exhibited
enough regularity to make the construction of a general (business cycle) theory possible.
The main objective of Lucas’ work was to show that, unlike the keynesian view holding that
business cycles could only be the result of disequilibrium, an equilibrium theory of economic
fluctuations was indeed feasible.
4.1.2
Theory and method
Lucas had a precise view of what macroeconomics ought to be. This view can be summarized
in the following points:10
1. There should be no distinction between the principles of microeconomics and those
of macroeconomics. Hence macroeconomics should be built on (choice-theoretical)
microfoundations.
2. Macroeconomics is based on general equilibrium analysis, accounting for the behavior
of aggregates as a result of the interactions between the different parts of the system
(agents and firms interact through markets and prices).
3. Macroeconomics is dynamic, i.e., dealing with the economy over time. As a result,
agents’ expectations about the future must play a crucial role.
4. Uncertainty plays a key role, by ascribing probabilities to future states of the world
and to the occurrence of unexpected shocks.
10
De Vroey (2018).
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5. There is a coincidence between a (macroeconomic) theory and a model. This view
broke with a tradition in social sciences that would typically conceive a theory as a
more general construct composed of a more general set of (qualitative) propositions.
Lucas postulated the exact coincidence between a model and a theory.
6. A model (i.e., theory) is intrinsically non-realistic. According to Lucas, insistence
on the realism of an economic model “subverts its potential usefulness for thinking
about reality.” Put differently, a “good model” will not be exactly more “real” than a
poor model, but will provide a better imitation of reality. The central assumptions of
macroeconomic models, such as rational expectations, ought to be viewed as modeling
devices, rather than propositions about reality: “One can ask whether expectations are
rational in a particular model of the United States economy; one cannot ask whether
people in the United States have rational expectations.” Lucas insisted on the fact that
the central question in macroeconomics is which necessarily abstract models can help
to answer practical questions of economic policy. Macroeconomics models are fictitious
economies, manipulating which we can learn about the functioning of real aggregate
economies.
Economics and anthropology. Concerning his view of what models are and what
role they should play, Lucas developed an insightful analogy between economics and anthropology. In his view, economic theory, like anthropology, works by studying societies
which are in some relevant sense simpler or more primitive than our own, “in the hope that
relations that are important but hidden in our society will be laid bare in simpler ones.”
Unlike anthropologists, however, economists simply invent the primitive societies they study.
This method of society-invention is the source, according to Lucas, of the utopian nature of
(macro)economics, and of the “mix of distrust and envy with which economists are viewed
by fellow social scientists.” The point of studying wholly fictional, rather than actual, societies (economies) is that it is relatively inexpensive to subject them to external forces (i.e.,
exogenous disturbances) and study how they react. “If, subjected to forces similar to those
acting on actual societies, the artificial society reacts in a similar way, we gain confidence
that there are usable connections between the invented society and the one we really care
about.”11
11
De Vroey (2018, pag.179.
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4.1.3
Equilibrium concept
The traditional (keynesian macroeconomics) view of equilibrium is that it means a state of
rest, in analogy to a pendulum. When we observe that the pendulum is moving we can infer
that disequilibrium exists. Hence disequilibrium happens all the time, in that we observe
prices and quantities in markets changing continuously. According to the traditional view,
the economy would be generally out of equilibrium, with re-equilibrating forces acting to
restore the pendulum in the position of rest.
In a famous paper entitled “After Keynesian Macroeconomics” (1979), Lucas and Sargent proposed a radically different concept of macroeconomic equilibrium. A model economy
being in equilibrium means that in each point in time two conditions are met:
1. Markets clear, i.e., prices adjust to make supply and demand coincide in each market.
2. Agents display an optimizing behavior (through intertemporal planning), given their
budget constraints, their preferences (i.e., households’ utility), and technology (i.e.,
firms’ production function). In this view, “equilibrium” (i) prevails all the time; (ii)
refers only to model economies, not to real ones. As such, equilibrium ought to be
understood as a feature of the way in which economists look at reality, rather than as
a characteristic of reality. According to Lucas, the mere notion of disequilibrium had
to be abandoned, because it referred to “unintelligent behavior,” i.e., a behavior not
grounded in rigorous microeconomic foundations.
This concept of equilibrium laid the ground for a profound transformation in macroeconomics and business cycle theory. If macroeconomic facts such as aggregate fluctuations have
to be interpreted through the lens of households’ and firms’ optimizing behavior, recessions
cannot be interpreted as mistakes, or pathologies, but rather as the result of efficient choices.
This view of aggregate fluctuations later informed the Real Business Cycle paradigm, initiated by authors (and also Nobel prize winners) F. Kydland and E. Prescott.
5
Rational expectations and economic policy
In the late 1960s and early 1970s scholars such as R. Lucas, T. Sargent, and N. Wallace,
based on the previous analysis of Muth (1960), began thinking about a deeper theory of how
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expectations are formed. They developed the logical and formal apparatus of rational expectations (RE). This methodological breach turned out to have deep implications economists’
view of the role of economic policy.
Expectations are rational when agents make the best forecasts they can with the available information. In other words, when the agents fully incorporate all the available information in predicting future values of relevant variables. It is important to notice that
those forecasts do not need to be necessarily correct. Hence observing forecasting errors by
individuals does not constitute evidence against rational expectations. Instead, RE imply
that agents do not systematically make forecasting errors. Thus agents’ expectations may
turn out to be wrong, but must be correct on average over time. Put differently, although
the future is not fully predictable, agents’ expectations are assumed to be unbiased. This
is because, conditional on all the available information (which could well be imperfect, due
different types of frictions), the agents make the best possible predictions. Under RE agents
can make “mistakes,” but cannot make systematic mistakes, i.e., they cannot be surprised
systematically.
Definition. Let’s proceed a bit more formally. Let’s define Et yt+1 as the mathematical
expectation held at time t of variable yt+1 . In order to define this mathematical expectation
Et (·) as rational, we will make two further assumptions.
1. Model. All agents know the model according to which the behavior of variable y over
time is determined. For instance, the behavior of variable y could obey the following
equation linking the current value of yt to its future expected value and to the current
value of another variable, xt
yt = αEt yt+1 + γxt
(7)
where α and γ are parameters. Hence all agents in the economy know that equation (7)
is the model driving the behavior of y over time, including parameters α and γ. In many
instances in the real world it is obviously unrealistic to assume that all agents know the
model of the economy from the very beginning. Individuals could be learning over time, and
could disagree in their beliefs about the future value of y.12
12
Learning and belief disagreement (or dispersion) are integral part of many current models in modern
macroeconomics 2.0.
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2. Information set. All agents are assumed to have the same information set at time t.
Let Ωt be the set of available information at time t including current and past values
of variables y, x and z:
Ωt = {yt−j , xt−j , zt−j } j = 0, 1, 2, ...∞
where z is any other variable outside the model that could be useful in predicting the
future value of y.
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Business Cycles, Imperfect Information,
and the Effects of Money on Output
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6
Business Cycles: from disequilibrium to equilibrium
analysis
One of the fundamental theme of macroeconomics concerns the causes of aggregate fluctuations, i.e., expansions and downturns in GDP, employment, consumption, investment, etc...
How can episodes such as the Great Depression of 1929, or the Great Recession of 2008,
even happen? What are the implications of large recessions for aggregate income, prices,
(un)employment and wages?13 Against this backdrop, a key sets of questions relate to the
role of policy. What is the effect of monetary policy on real variables such as output and
employment? How can monetary and fiscal policy be designed in order to prevent aggregate
downturns? And if downturns actually happen, how should those policies be designed?
The following sections will develop a more formal analysis. As clarified above, and since
the work of Keynes, the field of macroeconomics has revolved around three fundamental
questions.
• What determines aggregate business cycles fluctuations?
• What is the role of (monetary) policy in determining those fluctuations?
• What primitive elements can generate a Phillips curve (PC)?
We will study these questions through the lens of the scientific paradigm shift from the
Keynesian disequilibrium view of how the economy works to the equilibrium view of modern
macroeconomics, pioneered by the work of Robert Lucas. Central to our analysis will be the
islands model developed by Lucas (1973).14 Among other important contributions, Lucas
was the first, within that model, to show how to derive a Phillips curve conditional on the
discipline imposed by rational expectations. The island model is well worth studying because
it turned out to have crucial methodological implications in macroeconomics.
13
These questions concern the branch of macroeconomics that studies short-run aggregate fluctuations.
Obviously the complement of these questions includes the ones related to the determinants of long-run
growth. These notes focus entirely on the issue of short-run fluctuations.
14
The island model is developed by Lucas in “Some International Evidence on Output-Inflation Tradeoff,”
American Economic Review (1973). Another key reference is the 1975 seminal paper “Expectations and the
Neutrality of Money .”
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6.1
Classification of variables
Before plunging into any formal analysis, it is useful to introduce a classification of variables
into three different categories: respectively endogenous, “taken as given,” exogenous.
1. A variable xj is classified as endogenous when it is chosen by agent j (a consumer,
a firm,..) whose individual decision .problem is under analysis. The value of xj is
therefore determined within the equilibrium of the model.
2. A variable xj is taken as given by individual agent j when agent j’s individual decision
on other variables (yj ,zj ,..) ̸= xj ... does not affect the equilibrium value of xj .
3. A variable x is classified as exogenous when its value is determined outside the equilibrium of the model.
It is worth noting the difference between (2) and (3). Logically the relationship can be
described follows:
exogenous
→
“taken as given”
↚
If a variable is exogenous, then it is necessarily taken as given in the maximization problem
of any individual agent; but the reverse is not true. A variable can be “taken as given”
(or: treated parametrically) by an individual agent because she is atomistic in a given
market: for instance, a price-taker firm in a perfectly competitive market. But this does
not necessarily mean that the same variable is exogenous in the (general) equilibrium of
the economy. Continuing with the price example, consider a perfectly competitive market
populated by a large number of firms all producing the same good. While the price is taken
as given from the viewpoint of an individual firm, the equilibrium value of the good’s price
will be determined (and therefore is ultimately an endogenous variable) by the balance of
aggregate supply and aggregate demand in that given market.
6.2
Imperfect information and rational expectations
Recall our three central questions: What generates business cycles? What primitive elements can generate a Phillips curve (PC)? What is the effect of money on output? Lucas
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(1973) presented the first theoretically satisfactory answer to these questions. His theory is
based on the combination of two elements: imperfect information and rational expectations.
To preview, these primitive elements can generate three main results: (i) business cycle
fluctuations due to the agents’ misperception of individual vs aggregate shocks; (ii) a shortrun upward sloping Phillips Curve; (iii) real effects of money on output, in particular the
difference (discussed at length above) between the effects of anticipated vs. unanticipated
variations in the money supply.
In Lucas’ model agents have imperfect information and cannot unambiguously distinguish whether a local price increase is due to rising demand for their own product or a
general increase in the price level because of, for instance, an aggregate expansion of the
money supply (which would involve all islands). In contrast to previous (Keynesian) disequilibrium analysis, this is the first example of consistent equilibrium analysis in the sense
that all important variables are determined within the model (as opposed to being exogenously assumed), and that agents feature the discipline of rational expectations, i.e., they
can make forecasting errors, but those errors cannot be systematic.
6.3
The islands model
Consider an aggregate economy where there exist Z islands (or markets), each indexed by
z. On each island there is a producer who charges price pt (z), where z denotes a particular
island. We shall denote the aggregate price as pt , which is simply the average of p(z) across
all Z islands.
Individual producer supply. Supply (and production) in island z, yt (z), is determined by expected relative prices; when producers expect a high relative price of the good
produced on their island, they produce more of it.15 However, supply decisions are made
based on imperfect information. The nominal price of the good produced on each island
is observed only on that island, and the aggregate price level is observed only with a lag.
Hence the individual producer supply curve reads:
yt (z) = γ · [pt (z) − E (pt | It (z))] , γ > 0
15
(8)
One could think of the underlying market structure as one of perfect competition. Across all islands all
producers produce the same identical good, and therefore each producer z is price-taker.
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where It (z) denotes the information set available to producer in island z up to time t − 1 and
including the price pt (z). In the above expression, E (pt | It (z)) denotes the mathematical
expectation of the aggregate price level pt held by producer z conditional on the information
set It (z). All variables above are in logs and should be thought of as deviations from their
respective trend (or mean) values. Equation (8) embodies the idea that each individual
supplier increases production only in response to unexpected variations in its own price
relative to the aggregate price level. No distinction is made between workers and producers.
Hence, in this model, each firm z is like a small farmer (who is simultaneously producer and
worker) who is induced to produce more by an unexpected increase in the actual price.
Imperfect information. Suppliers do not observe the aggregate price level directly.
Hence they must form an estimate of it. Before entering the market, each supplier z has
a prior distribution for pt , which is normal, with mean E (pt | It ) and variance σp2 , where
It denotes the information set consisting of full information about the economy up to and
including time t−1, but not including the individual price pt (z). Information barriers prevent
each producer from knowing whether prices have gone up simultaneously in all markets or,
else, only in its own market. Since producer z does not observe the aggregate (or average)
price level, he/she has no information about whether a change in p(z) is also a change in the
relative price of its own good.
Rational expectations. We assume that each supplier has rational expectations
(RE). As clarified above, what RE mean is that economic agents employ all relevant information in formulating their forecasts. In this context, this means that each supplier
knows the distribution of pt (and therefore its mean and variance). A crucial corollary of RE
is that agents cannot make systematic forecast errors. Hence assuming RE in our context
implies
pt = E (pt | It ) + ϵt
(9)
where ϵt is a forecast error with mean zero, E (ϵt ) = 0, and variance E (ϵ2t ) = σϵ2 . Equation
(9) implies that, on average, the actual aggregate price level coincides with its expected
value. In other words, agents cannot make systematic errors in forming expectations about
the aggregate price level. Notice that (9) implies
V ar(pt ) = σp2 = E [pt − E (pt | It )]2 = σϵ2
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Individual and aggregate prices. The price in each island pt (z) is assumed to differ
randomly from the aggregate price level pt :
pt (z) = pt + θz,t , θz,t ∼ N (0, σθ2 )
(10)
where θz,t is an in independent, identically distributed (i.i.d.) disturbance with normal
distribution, mean zero and variance σθ2z . Hence θz,t is an island-specific shock. Combining
(9) and (10) we can write:
pt (z) = E (pt | It ) + (ϵt + θz,t )
| {z }
(11)
ηt
Hence in every period t the island-specific price deviates from the expected (or estimated)
average price by a composite random factor ηt . If the individual producer had perfect
information about the aggregate price level, then it would respond only to the island-specific
disturbance θz,t :
ϵt = 0 → pt = E (pt | It )
which implies
pt (z) = pt + θz,t
(12)
Due to imperfect information, however, each individual producer observes the gap ηt between
its own price pt (z) and the expected aggregate price level. As a result producer z cannot
distinguish whether the gap ηt is due to the aggregate forecast error ϵt or to the idiosyncratic
disturbance θz,t . Under imperfect information, each producer only observes the composite
forecast error ηt ≡ ϵt + θz,t .
Signal extraction. Each individual producer therefore faces a signal extraction problem, i.e., it needs to decide how much of the composite error is due to mistakes in forecasting
the aggregate price level (ϵ) and how much is due to the relative price shock (θz,t ), and to
only alter output in response to the latter. This is called a signal extraction problem. How
is it solved? It seems natural to assume that each individual producer cannot observe the
time-t current values of ϵ and θz , but can observe their historical data at time t − 1, t − 2,...
Hence the problem for the individual producer z is to guess which proportion of the variability of the idiosyncratic disturbance θz,t is due to the composite error ηt . In other words,
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the producer needs to understand how much of the variability of θz,t needs to be “extracted”
from the composite error ηt . This can be done by running the following OLS regression:
θz,t = β · ηt + ut
(13)
where the constant term in the regression is assumed to be zero (given that all disturbance
terms have mean zero), and ut is a normally distributed error with mean zero and variance
σu2 .
The expression for the estimated value of β is given by the standard OLS formula:
Cov(θz,t , ηt )
βb =
V ar(ηt )
(14)
From (13), the expected value of θz,t conditional on ηt can therefore be written:
b t
E (θz,t | ηt ) = βη
(15)
Let’s compute V ar(ηt ) first. We can write:
V ar(ηt ) = V ar(θz,t ) + V ar(ϵt )
= σθ2z + σϵ2
We turn to the covariance term next. We write:

 







θz,t − Eθz,t · (θz,t + ϵt ) − E(θz,t + ϵt )
Cov(θz,t ,ηt ) = E
|{z}
| {z } 

=0
=0
= E {θz,t · (θz,t + ϵt )}
2
= E θz,t
+ θz,t ϵt
2
= Eθz,t
+ E (θz,t ϵt )
| {z }
=0
=
σθ2z
Notice that Eθz,t ϵt = 0 holds because the two error terms, θz,t and ϵt are independent. The
final expression for βb therefore reads:
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βb =
σθ2z
σϵ2 + σθ2z
(16)
Each individual producer’s best guess is that θz,t is a fraction βb of the composite error term
ηt (which is observable). In other words, the best guess of θz,t is its conditional expected
value, as from (15), βb · ηt , where βb is given by (16) and σ 2 and σ 2 are computed based on
ϵ
θz
the observed historical data on θz,t and ϵt .
Forecast of aggregate price level. We can now take stock of our results and write
an expression for an agent’s best guess on the aggregate price level. The latter is given, from
(10) by:
pt = pt (z) − θz,t
Taking expectations of the above expression conditional on It (z), and therefore conditional
also on ηt :
E {pt | It (z)} =
pt (z)
| {z }
−E {θz,t | It (z)}
(17)
known at time t
= pt (z) − βb · (θz,t + ϵt )
= pt (z) − βb · [pt (z) − E (pt | It )]
b t (z) + βE
b (pt | It )
= (1 − β)p
where the second to last step follows from equation (11). Hence each individual agent’s
forecast of the aggregate price level is a weighted average of the current individual price
observed in the market and the prior forecast on the same aggregate price level (based on
information only available at the end of time t − 1).
Bayesian updating. In terms of Bayesian statistics, the one in (17) corresponds to a
process of updating, whereby each agent z formulates a prior about the price level E (pt | It )
and updates it with new information after observing the individual price pt (z), thereby
forming the posterior forecast E {pt | It (z)}. The relative weights given to the prior and to
b
the new information are given respectively by βb and (1 − β).
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E (pt | It )
| {z
}
→ pt (z)
new
→ E {pt | It (z)}posterior
information
prior based
on information set
It
Aggregate supply curve. We can now substitute (17) into (8)
b t (z) − γ βE
b (pt | It )
yt (z) = γpt (z) − γ(1 − β)p
= γ βb · [pt (z) − E (pt | It )]
(18)
yt = γ βb [pt − E (pt | It )]
(19)
Aggregating across all islands z ∈ Z we obtain:
Equation (19) is the aggregate supply curve of the economy. It implies that aggregate
b to the unexpected variations of the
production responds positively (with an elasticity γ β)
aggregate price level, i.e., deviations of pt from the (prior) forecast E (pt | It ). This is Lucas’
explanation of the Phillips curve.
Rearranging, and then adding and subtracting pt−1 on both sides of (19), we can write
the PC in terms of the inflation rate:
−1
b
yt + [E (pt | It ) − pt−1 ]
pt − pt−1 = γ β
| {z }
inflation
rate
(20)
Hence the inflation rate (between t − 1 and t) depends on current economic activity
and on the expected variation of the general price level (conditional on information held at
the end of time t − 1) relative to the previous period price. Hence it is any unforecastable
(“surprise”) movement in inflation that generates movements in real economic activity.
6.3.1
Implications of Lucas aggregate supply curve
Equation (19), or (20) has a number of important implications.
1. It is an aggregate Phillips curve (i.e., a positive relationship between inflation and
output, or equivalently a negative relationship between inflation and unemployment)
derived from first principles, meaning from a description of the economy based on
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disaggregated markets. This result addresses the criticism of traditional Keynesian
macroeconomics, whereby aggregate relationships were simply postulated, and not
derived from microeconomic foundations.
2. Despite imposing the discipline of rational expectations, equation (19) describes a
Phillips curve. In particular, it suggests that economic activity in period t depends
positively on the surprise variations in the aggregate price level (we will see in a
shortwhile what are the forces that cause those variations in the economy, monetary
shocks in particular). As already hinted above, this result depends on imposing, along
with RE, also the assumption of imperfect information.
3. Equation (19) shows that a PC exists in the short run, but not in the long run. Since,
under rational expectations, agents cannot make systematic errors, it must be true
that on average (which is an equivalent way to express “the long run”) the forecast
of the aggregate price level is equal to the actual price level. In other words, in the
long-run, pt = E (pt | It ), which implies yt = 0, i.e., output always is line with its trend
value. Hence the model provides a theory of why, conditional on the discipline of RE,
a Phillips curve exists in the short run and not in the long run, consistent with the
original natural rate hypothesis of Friedman.
6.3.2
Demand side
So far we have only discussed the supply side of the model, we now turn to the demand side.
We assume that demand in each island, ytd (z), depends on nominal money and the price
level in each island,
ytd (z) = mt (z) − pt (z)
The money supply in each island depends on the aggregate money supply plus a random
term µz,t with mean zero and variance σξ2 .
mt (z) = mt + µz,t
Money supply. The aggregate money supply, controlled by the monetary authority,
is assumed to evolve according to:
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mt = mt−1 +
+µt
gm
|{z}
(21)
money
growth rate
where µt is the unexpected (random) component of money growth. Notice that, since µt has
mean zero, the above equation implies:
E (mt | It ) = mt−1 + gm
(22)
As a check, notice the unexpected (surprise component) of the money supply can be written:
mt − E (mt | It ) = µt
6.3.3
Equilibrium under rational expectations
Equilibrium in each island implies:
yt (z) = ytd (z)
(23)
γ βb [pt (z) − E (pt | It )] = mt (z) − pt (z)
{z
} |
{z
}
|
(24)
γ βb [pt − E (pt | It )] = mt − pt
(25)
which can be written:
supply in
island z
demand in
island z
Aggregating across island z ∈ Z we obtain:
Next, we once again apply the discipline of rational expectations. Taking conditional expectations of both sides of 25 we obtain:
γ βb · E








= E {(mt − pt ) | It }
[pt − E (pt | It )] | It
|
{z
}






forecast error
on aggregate price level
= 0
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(26)
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Notice that the application of rational expectations lies precisely in the implication
whereby the forecast error on the aggregate price level must on average be zero. The above
equation therefore implies:
E (pt | It ) = E (mt | It )
(27)
= mt−1 + gm
where the last expression follows from (22). Using the assumption of rational expectations
we have therefore found a solution for the expectations of the aggregate price level.
Equation (27) implies that the expected aggregate price level (conditional on information
available at end of t − 1, or equivalently beginning of time t) is equal to the expected level of
the money supply. In turn the latter is given by the past level of the money supply (which
is known) and a constant growth rate (which is known, and exogenous, because it is set by
the monetary policy authority).
Solutions. Armed with (27) we can now find solutions for the remaining variables of
the model. Substituting E (pt | It ) into (24) we obtain:
Rearranging:
γ βb [pt (z) − mt−1 − gm ] = mt (z) − pt (z)
(28)
b t (z) = mt + µz,t +γ βm
b t−1 + γ βg
b m
(1 + γ β)p
| {z }
(29)
mt (z)
Finally, substituting for mt from (21) we obtain the market clearing price for each island
(market):


1

pt (z) = (mt−1 + gm ) +
b 
|
{z
} (1 + γ β)
E(mt |It )
µz,t
|{z}
idiosyncratic
shock

+

µt 
|{z} 
(30)
aggregate
shock
Hence island z’s price level depends on the expected aggregate quantity of money, and on
both the aggregate and idiosyncratic money shocks (µz,t and µt respectively).
Aggregating across islands we obtain the general price level in the economy:
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pt = (mt−1 + gm ) +
µt
b
(1 + γ β)
(31)
Hence the aggregate price level depends on two terms. The first term (mt−1 + gm ) =
E (mt | It ) is the expected quantity of money. The second component is the unexpected
forecast error in the money supply, µt .
Only unanticipated money affects output. Finally, we can derive an expression
for the equilibrium level of aggregate output. Substituting (31) into (19), and using (27) we
can write:
b t − γ βE
b (mt | It )
yt = γ βp
(32)
γ βb µt
b (mt | It )
− γ βE
= γ βb (mt−1 + gm ) +
b
(1 + γ β)
!
γ βb
=
· µt
b
(1 + γ β)
The important result in (32) is that the equilibrium level of aggregate output (unlike the price
level) depends only on the unexpected component of the quantity of money. Equivalently,
it is only the unanticipated component of monetary policy that matters for fluctuations in
output. The anticipated component of money is completely neutral. Once again this is the
formalization of the Friedman-Phelps idea, now established within the context of a consistent
theory based on equilibrium and rational expectations.
Model-consistent expectations. Equation (31), which is the equilibrium solution
for the aggregate price level, shows a key property of rational expectations: namely, that
agents’ expectations are model consistent. To understand this point, consider the logical steps we have adopted in deriving our solution. We have first assumed that in the
model E (mt | It ) = (mt−1 + gm ). Under rational expectations we have pointed out that
E (mt | It ) = E (pt | It ) and therefore
E (pt | It ) = (mt−1 + gm )
(33)
Equation (31) allows us to verify whether agents’ expectations are consistent with the model.
By taking conditional expectations of (31) we obtain
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E (pt | It ) = (mt−1 + gm ) +
1
E (µt | It )
b | {z
}
(1 + γ β)
(34)
zero
= (mt−1 + gm )
which corresponds exactly to (33). This feature of internal consistency between the model
and the agents’ expectations is a key property of rational expectations, and is a crucial
methodological innovation that has had profound implications on modern macroeconomics.
Signal extraction parameter and structural shocks. From equation (11) recall
that in the initial specification of the model we assumed the existence of both an idiosyncratic
and aggregate disturbance, θz,t and ϵt respectively. The expression (30) for the equilibrium
price in each island now reveals that, in terms of underlying monetary disturbances, those
error terms are respectively given by
θz,t =
µz,t
(35)
b
(1 + γ β)
µt
(36)
b
(1 + γ β)
The underlying monetary disturbances µz,t and µt are therefore the true (so-called strucϵt =
tural) shocks of the economy. The observed disturbances θz,t and ϵt are a parameter convolution of the structural shocks. In the terminology of econometricians the latter are defined
as reduced-form residuals.
Lucas’ critique of policy evaluation. We can also derive an expression for the
signal extraction parameter:
βb =
σθ2z
σϵ2 + σθ2z
1
b
(1+γ β)
2
= =
=
(37)
2
σµ2 z
1
σµ2 + σµ2 z
b
(1+γ β)
σµ2 z
σµ2 + σµ2 z
1
σ2
1 + σ2µ
µz
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The signal extraction parameter βb is therefore a function of the relative variability of
idiosyncratic and aggregate monetary shocks. The higher the variance of island-specific
b and therefore the lower the weight assigned
monetary shocks σ 2 the higher the value of β,
µz
to pt (z) in each agent’s updating process summarized by (17).
The fact that the “Bayesian updating parameter” βb depends on the variability of the
monetary shocks is at the heart of another key implication of Lucas’ theory: the so-called
econometric policy evaluation critique. The critique starts from acknowledging that parameters that characterize the individual agents’ decision making process (i.e., βb in this case)
cannot be considered as structural, in the sense of being invariant to changes in government
policy variables (monetary policy in this case). This creates a key challenge for applied
macroeconomists. The aggregate supply function (19) provides an example. Suppose, in
the most favorable scenario, that the expectations term E (pt | It ) were observable. Then,
naively, one could think of estimating the “slope” of the Phillips curve equation (19) (i.e.,
the elasticity of output variations to price level variations) by running a regression of yt on
the price level pt (as well as on the observable expectations term):
yt = b0 + b1 pt + ..
In running the above regression the assumption would be that the coefficient of interest
linking prices and output, b1 , is a constant, i.e., it is time-invariant. But the theory unb cannot be
derlying (19) and (37) shows that the parameter linking those two variables, β,
considered invariant, and in particular cannot be considered invariant to policy. From (37),
in fact, βb is a function of the volatility of the monetary shocks σ 2 and σ 2 , which is under
µ
µz
the control of monetary policy. In principle, that volatility could change over time, because
the monetary policy regime could change. For instance, because a central bank, under a new
governor, shifts from being passive to being particularly active in using the money supply
instrument. The so-called “Lucas critique” flags a critical warning in interpreting the estimated coefficients of regressions among aggregate variables (output and price level in this
case) as structural, i.e., truly invariant to policy changes.
6.4
Lessons from Lucas theory
There are a number of long-lasting lessons that can be derived from the Lucas island model,
which make it one of the founding pillars of modern macroeconomics.
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1. Phelps (1970) first developed the idea that if markets are decentralized and agents have
only limited information about their markets, the same agents would end up responding
to shocks they would not have responded to, had they had perfect information. Lucas
however was the first one to develop this idea into a formal and consistent model.
2. Even more importantly, Lucas formalized, and obtained as an equilibrium implication, the Friedman-Phelps idea whereby any attempt to exploit the Phillips curve
relationship between prices and output by more expansionary monetary policy, and
permanently increase output and employment, would be fruitless and only result in
more inflation. Lucas’ theory is therefore the first one in which, despite the presence
of rational expectations, short run deviations between inflation π and inflation expectations π e (and therefore between u and un , see equation (6)) can actually take place.
This possibility is due precisely to the introduction of informational frictions. Suppose
in fact that the model only assumed rational expectations. A strict application of
the rational expectations logic to the generic PC equation (6) would lead to conclude
that a tradeoff between inflation and unemployment, let alone in the long run, does
not even exist in the short run. For under rational expectations, strictly speaking,
and unlike the view of Friedman, inflation expectations would be extremely quick to
adjust, keeping inflation in line with its expected value in each period, and therefore
unemployment (or output) in line with its natural rate. Lucas therefore added another ingredient to rational expectations: imperfect information. Due to the presence
of imperfect information, it is rational for the producers in the model to interpret a
proportion of each price increase as caused by increased demand for its own product,
and therefore to increase output somewhat. The implied positive relationship between
prices and output describes a Phillips curve.
3. An important consequence of Lucas’ theory is that it highlights, and derives from first
principles, the crucial distinction between anticipated and unanticipated changes in
monetary policy.16 If changes in the money supply, and the resulting changes in the
aggregate price level, are anticipated, then each “islander” is not misled by any price
changes that it observes. In other words, it will correctly interpret the price change in
16
In his Nobel price lecture Lucas emphasizes the distinction between the effects of anticipated and unanticipated changes in monetary policy as the key idea behind the birth of modern macroeconomics in the
1970s. See here https://www.nobelprize.org/prizes/economics/1995/lucas/lecture/
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its own island as the result of an aggregate (and not idiosyncratic) change in the price
level. Consequently, it will not adjust production, and the neutrality of money will
occur even in the short-run. With unanticipated changes in the money supply (and
inflation), the islander faces an imperfect information problem, and will partly adjust
production. Therefore, monetary policy can affect output only as long as it surprises
individuals and firms in an economy, with this result holding despite the presence of
rational expectations.
4. With the so-called “Lucas critique” the model flags a warning, for the first time, about
the possibility of interpreting the coefficients of statistical regressions among aggregate
variables as truly structural, i.e., invariant to policy changes. This is exemplified by our
central result whereby the Bayesian updating coefficient βb turns out to be a function
of the underlying behavior of monetary policy. To put it simply, that means that
b is affected by how policy is
the individual decision-making process (summarized by β)
conducted. Since policy behavior can change over time (because of regime shifts: for
example, moving from flexible to fixed exchange rates, adopting a regime of inflation
targeting, etc..) it would be fallacious to consider the individual behavioral parameters
b also time invariant.
(i.e., β)
5. Finally, one of the long-lasting impact of Lucas model is methodological, for several
reasons. It illustrates the power of rational expectations as a modelling technique.
It shows how an equilibrium can be derived, and how agents’ expectations at the
equilibrium can be consistent with the underlying primitive elements of the economy.
Also, it shows how, by exploiting the combination of imperfect information and rational
expectations, to use a simple general equilibrium model to derive a structural model
of economic fluctuations, i.e., a model where, at the equilibrium, prices and quantities
respond to the primitive shocks (in Lucas’ case monetary shocks) specified in the model.
6.4.1
Limitations and critiques
Lucas’ island model, despite being a pioneer, was also subject to a number of criticisms.
1. Absence of decision-making problem. The first criticism is that, in each island z, the
price p(z) of each individual producer is not derived from the solution of any profit
maximization problem. This point begs the question: how are the price p(z) and the
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quantity produced y(z) set in each island? What is the underlying decision-making
problem? What is the underlying market structure (perfect competition, monopolistic
competition, or else)? In a nutshell, Lucas’ model, despite being a quantum leap
relative to previous Keynesian theory, still featured a problem of not fully satisfactory
microfoundations.
2. Persistence problem. As a description of the economy, the islands model has no inherent
dynamics. In other words, unanticipated monetary shocks, although not neutral on
output, affect the latter only for one period. There is now ample empirical evidence that
monetary shocks have long lasting effects on real variables such as output, consumption,
and investment. Reproducing this evidence is however a challenge even for the most
recent models in monetary economics, a problem usually defined as the “persistence
problem.” More generally, the information frictions in the Lucas model are often viewed
as too short lived to be able to explain both the observed depth and persistence of
aggregate fluctuations in GDP and other real variables.
3. Monetary shocks only? The assumed sources of exogenous disturbances in the islands
model are monetary disturbances (whether idiosyncratic or aggregate). This is because Lucas was interested in studying the differential effects on output of anticipated
vs unanticipated monetary shocks. Lucas’ theory of business cycle fluctuations is indeed based on the misperception of monetary shocks. Is this view empirically relevant?
Are monetary shocks truly the relevant one in determining business cycles? In the early
1980s a new line of research, labeled Real Business Cycle (RBC) theory developed the
view that business cycles are caused by real shocks such as shocks to total factor productivity (i.e., exogenous shifts in the production possibility frontier of the economy).
RBC theory kept the rigor of general equilibrium and rational expectations, but refrained from assuming informational frictions. Advancing relative to Lucas work, RBC
models put instead central emphasis on the role of dynamics, microfoundations and
intertemporal substitution.
4. Interpretation of imperfect information. It is unclear how imperfect information should
be interpreted in equilibrium models. Is Lucas model to be interpreted literally, or
should it be considered as an example of coordination problems that are generally
pervasive in large disaggregated economies? How can agents suffer of a problem of
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coordination of information about average prices and/or quantity of money, when this
information are publicly available at an increasingly lower cost? The answer could be
that, despite this information being available, agents choose rationally to ignore it.
This idea paved the way to a more recent school of thought called rational inattention.
Developments after Lucas. Despite the above criticisms, Lucas’ islands model was
key to the subsequent developments in macroeconomic theory. Research after Lucas’ islands
model abandoned the idea of imperfect information, and took two complementary avenues.
The first avenue of research is the above-cited RBC theory, as exemplified by the pioneering
work of Kydland and Prescott (1983) and King and Plosser (1982). The second, is the so
called New Keynesian theory, i.e., a theory of business cycles and monetary policy based on
nominal rigidities (in prices, wages or both), however still holding the discipline of rational
expectations.17 Interestingly, the “imperfect information” view is currently having a strong
resurgence of interest in macroeconomics, especially disputing the view that expectations
are actually rational and completely homogenous across agents. As hinted above, a recent
line of research, called rational inattention, builds around the role of imperfect information.
The rational inattention view emphasizes agents’ limited capacity of processing information,
despite that information being publicly available.
7
Modern macroeconomics in brief
Taking stock of what we have learnt so far, the main pillars of modern macroeconomics can
be summarized as follows:
1. Dynamics;
2. General equilibrium;
3. Microfoundations.
Modern macroeconomics is dynamic, i.e., it places the intertemporal dimension of agents’
(firms or households) decisions at center-stage. Think, for instance, about the key role played
by expectations in financial markets, as well as in firms’ price and wage setting decisions;
or, alternatively, about the determinants of households’ consumption-saving decisions: will
17
We will introduce the New Keynesian theory in a later section.
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an agent increase consumption by more in case of a lottery win or in case she finds a new
job? Or else: if the government aims at balancing its budget over time, and not period by
period, does a tax cut produce a rise in consumption, or simply in savings? These questions
can only be understood in a setting where the future matters as much as the present.
The second pillar is general equilibrium. That means viewing the aggregate economy
as one where all markets interact simultaneously, through the adjustment of prices. Thus
individual agents make their decisions from their own perspective by taking prices (of any
kind: goods prices, wages, interest rates) as given, but those prices are actually determined
through the balancing of supply and demand in each market.
The third pillar is microfoundations. In modern macroeconomics, the analysis builds
from the decision-making problem of individual agents. Households take decisions in order
to maximize their utility, while firms in order to maximize profits, respectively subject to a
budget and a technological constraint. Macroeconomics studies how the interaction of these
optimizing individual decisions produce specific aggregate outcomes, such as unemployment,
inflation, or a financial crisis.
7.1
Distinction from IS-LM macroeconomics
All the above elements sharply distinguish (the analytical tools of) modern macroeconomics
from the introductory version based on the well-known IS-LM model, which was the central
apparatus of the so called “keynesian macroeconomics.” We can think of at least four reasons:
1. The IS-LM model is static, i.e., it does not assign any role to expectations about the
future. For instance, it is not logically possible, in that setup, to study the differential
effects on consumption of a temporary vs. a permanent reduction in wage income. If
a household looses its job, does it matter for her consumption decision whether he/she
expects this event to be purely temporary or permanent? These questions cannot be
even posed within the IS-LM model.
2. The IS-LM model does not view the economy as a system in which all markets interact
simultaneously and all prices are determined. Some prices, like good prices, are simply
fixed. Other prices, like nominal wages, are not even specified, because the underlying
market, the labor market, is not present.
3. The IS-LM model does not build its analysis on the (optimal) decisions taken by
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individual agents. Simply put, there are no economic agents specified in the model.
The logic in the IS-LM model (as in the traditional Keynesian view) is the one of
directly assuming the presence of aggregate quantities (consumption, income, etc..)
and to postulate the relationship between those variables. For instance, aggregate
consumption is a linear and increasing function of aggregate disposable income; or,
the aggregate demand of money is a decreasing function of the interest rate. These
relationships are simply assumed, and not derived from the decision-making problem
of individual agents.
4. The role of (monetary and fiscal) policy built into the IS-LM framework does not capture key aspects of economic policy itself. For one, the role of systematic policy, i.e.,
policy meant as a tool to stabilize the economy in response to underlying disturbances
(e.g., oil shocks, or financial or demand shock). In the IS-LM model economic policy
merely acts as an exogenous shifter of the money supply or of government spending.
Furthermore, and most importantly, the IS-LM model is not suited for the normative
analysis of policy: how should monetary policy respond to an oil shock? Should monetary policy react to a fall in asset values? Should the government expand purchases
when the economy plunges in a recession? Does the answer to the latter question depend on the reason for which the economy is in a recession? Modern macroeconomics
allows to provide a first answer to all these questions.
Example: taxes and consumption. What is the effect on consumption of a cut
in taxes? Answering this question (among many possible ones) helps to understand the
difference between modern macroeconomics (in which agents make decisions and both the
present and the future play a role) and the traditional IS-LM model. To make things simple,
we will only consider the case of so-called “lump-sum” taxes, i.e., taxes that are equivalent
to mere negative tranfers. Let’s wear the lens of the traditional IS-LM model first. In that
model, aggregate consumption is assumed to be a linear function of aggregate disposable
income:
C = c0 + c1 (Y − T )
(38)
where c0 and c1 are positive parameters (with c1 being the so-called marginal propensity
to consume out of income). Hence, in the IS-LM model, a fall in T unambiguosly raises
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disposable income, and therefore consumption. The consumption function described in (38)
is typical of the approach followed in IS-LM macroeconomics. All relationships are both
specified in terms of aggregates and postulated, i.e., not derived from primitive microfoundations.18 Suppose, on the other hand, that the world is made of individual consumers who
take decisions, and simply acknowledge that the future exists. If the government cut taxes,
and aims at keeping the level of government spending constant (i.e., not cutting spending
on roads, hospitals, public wages, etc..), it will be forced to increase public debt. In turn,
the higher debt will have to be repaid sometimes in the future. An individual agent, rationally, might incorporate this logic into his/her decision today. By anticipating that lower
taxes today will most likely mean higher taxes in the future, the agent might well decide
to save at least part of the reduction in taxes. In the case in which the agent decided to
save all the additional income stemming from lower taxes, the effect on consumption of the
reduction in taxes would be nil.19 We learn an important lesson from this example. Namely,
that by explicitly including a role for individual agents who take (rational) decisions, and
by acknowledging the mere presence of the future as well as of the past, we can provide a
radically different answer to the same question. While, in the IS-LM world, consumption can
only rise in response to a cut in taxes, in the alternative world of modern macroeconomics
the effect on consumption of the same tax cut might outright be zero.
18
As an alternative example, think of the “investment function” in the IS-LM model, whereby aggregate
investment is assumed to be a direct function of aggregate income and an inverse function of the interest
rate.
19
We will see later that this extreme result will constitute the core of the so-called Ricardian Equivalence
theorem.
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The Intertemporal Consumption-Saving Model
Consumption and Saving, Expectations, Ricardian Equivalence, Fiscal Policy,
Openness and the Current Account
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8
The intertemporal view
In this section we introduce an intertemporal model of consumption and saving. Relative
to the islands model presented in the previous section, we make the following modifications.
First, we remove the hypothesis of uncertainty and imperfect information, and study a
baseline setup with certainty. Second, we introduce an intertemporal dimension, i.e., a
distinction between a “present” and a “future.” Third, we assume an endowment economy,
i,e., one where the supply of goods is like “manna from heaven.” Fourth, we assume that
the economy is populated by a representative household, who consumes and saves in a real
financial asset. Fifth, we specify deeper microfoundations for the household problem. We
assign the household an objective function to maximize, its own intertemporal utility, subject
to a budget constraint. Finally, we assume that the economy is completely real, and that
money is absent.
9
A two-period model of consumption and saving
We introduce a two-period endowment model of consumption and saving. Conceptually, we
have two main goals, First, to illustrate the role of the intertemporal dimension in economic
decision-making. This pertains questions such as: does household consumption respond
more to temporary or permanent variations in income? What is the effect of variation in
the real interest rate on savings? The second conceptual goal is to illustrate how, through
the competitive general equilibrium, individual decisions (about consumption, saving, labor
supply, etc..) shape the behavior of aggregate variables. The two-period model, in particular,
is best suited to study the role of fiscal policy. In this model, the predictions regarding the
economic effects of variations in taxes and government spending are radically different from
the ones in the IS-LM model. Most importantly, the two period intertemporal model will
allow to illustrate a key benchmark concept for the study of fiscal policy - the Ricardian
Equivalence theorem.
Structure of the model. The main elements of the economy are as follows:
1. This is an endowment economy: in other words, output is generated as “manna from
haven” (and not produced using either labor or capital).20
20
An “endowment” economy contrasts to a “production” economy where goods are produced employing
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2. In each period, a representative consumer decides whether to (i) consume her endowment or (ii) save in a one period asset and earn the return from saving tomorrow
(higher future consumption).
3. From its individual perspective, the consumer saves facing a given real interest rate.
What this means is that the decision is taken by an agent whose actions cannot affect
the market price of savings (i.e., the real interest rate).21 An alternative way to state
this assumption is that the credit market is perfectly competitive; therefore any agent
participating in the market is atomistic (or, equivalently, price-taker).
4. A government needs to finance a given stream of expenditures via lump-sum taxes.
These are taxes that are independent of the realized income of the agent (as would
proportional and/or progressive taxes). By assumption, lump-sum taxes do not affect
the agent’s decision at the margin (i.e., they do not affect its optimality conditions,
but only its budget constraint).
9.1
Budget constraints of the consumer
In period 1 the consumer faces the following budget constraint
C+
S
|{z}
saving/
bond purchases
=
Y
|{z}
−T
endowment
of output in period 1
where C is consumption, S is saving in one period bonds (or borrowing), Y is the endowment
of goods and T are lump-sum taxes (or transfers).
Note:
S > 0 → saving
S < 0 → borrowing
′
The budget constraint in period 2 reads (where a superscript denotes variables in period
2):
′
C =
′
Y
|{z}
+
endowment
of output in period 2
(1 + r)S
| {z }
−T
′
interest earned
on previous period
savings
factors of production (labor and/or capital).
21
We will learn, however, that in the general equilibrium the real interest rate is endogenous. Hence the
decision of the individual that we consider corresponds to a partial equilibrium perspective.
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Terminal condition. We impose the terminal condition whereby the consumer wishes
to finish period 2 neither with assets nor with debt. Hence we have:
′
S = 0.
9.2
(39)
Intertemporal budget constraint
By combining the period 1 and period 2 budget constraints we obtain a central object of our
analysis: the intertemporal (or present value) budget constraint. Start by rewriting S from
period 1 budget constraint:
S =Y −C
(40)
Next, substitute S in period 2 budget constraint and obtain, using the terminal condition
(39):
′
′
′
C = Y − T + (1 + r) [Y − C]
(41)
Dividing the above equation through by (1 + r) yields:
′
C+
′
′
C
Y
T
=Y +
−T −
1+r
1+r
1+r
(42)
At this stage, define lifetime wealth as:
′
′
T
Y
−T −
we ≡ Y +
1+r
1+r
Lifetime wealth corresponds to the present value of income net of taxes.
(43)
Finally obtain an expression for the intertemporal budget constraint:
′
C
C+
= |{z}
we
1
+
r
| {z }
PV
PV
consumption
(44)
income
The above equation states, quite intuitively, that the present value of consumption must be
equal to the present value of net income. This is the single fundamental budget constraint
faced by the individual consumer.
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Figure 11: Graphical representation of the intertemporal budget constraint
Graphical representation. Rewrite the intertemporal budget constraint (44) as:
′
C = (we − C)(1 + r)
which in turn implies:
′
C = we(1 + r) − (1 + r) C
| {z } | {z }
intercept
(45)
slope
The graphical representation of equation (45) is given in Figure 11: future consumption C
′
is reported on the vertical axis and current consumption C is reported on the horizontal axis.
The slope of the budget constraint line BA is the real interest rate (1+r).
Point E on the BA line is a combination of current and future consumption such that the
agent consumes her own endowment in every period. Hence the consumer is neither a lender
nor a borrower. Points to the right of E correspond to an arrangement such that C > Y − T .
Hence in the current period the agent wishes to consume more than its net income, and
therefore is labeled as “borrower” (she will therefore borrow and S < 0). Points on the
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budget line BA to the left of point E correspond to an arrangement such that C < (Y − T ):
the consumer is therefore a saver (S > 0).
9.3
Preferences
The agent has preferences over a bundle composed of current and future consumption. These
preferences are represented by the utility function:
′
U (C, C )
(46)
The function (46) is (i) increasing in both arguments, (ii) separable, and (iii) strictly concave.
Formally this is summarized in the following assumptions:
′
′
∂U (C, C )
∂U (C, C )
> 0;
>0
∂C
∂C ′
′
(47)
′
∂U (C, C )
∂U (C, C )
′
= UC (C);
= UC ′ (C )
′
∂C
∂C
(48)
′
′
∂ 2 U (C, C )
∂ 2 U (C, C )
< 0;
<0
∂C∂C
∂C ′ ∂C ′
(49)
Normal goods. We assume that both current and future consumption are normal
′
goods. This implies that, at any given real interest rate, both C and C increase with
′
income (either Y, or Y or both).
9.4
Consumer problem
For simplicity we henceforth abstract from the presence of the government. Therefore:
′
T =T =0
The consumer’s goal is to maximize utility subject to:
′
max U (C, C )
s.t.
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(50)
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C +S =Y −T
′
′
(51)
′
C = Y − T + (1 + r)S
(52)
According to our classification of variables, and from the perspective of the individual agent,
′
the triple (C, C , S) is endogenous; the real interest rate r is taken as given; and the tuple
′
(Y,Y ) is exogenous.
Separable preferences. A typical form of separable preferences is the following:
′
U (·) = u(C) + βu(C )
β ∈ [0, 1]
In this formulation β is a preference parameter which measures the degree of relative
impatience (the higher β, the higher the weight assigned to future consumption relative to
current consumption, therefore the more patient the agent).
Indifference curves. Preferences can be represented by a set of indifference curves,
depicted in Figure 12. Any given indifference curve represents combinations of current and
′
future consumption (C, C ) which keep the level of utility constant. Assumptions (47) and
(49) imply two main main features of the indifference curves
1. “More is better:” by increasing current consumption, future consumption, or both, the
level of utility increases. This property, which follows from (47), has two implications
for the shape of the indifference curves. First, each indifference curve is negatively
sloped. Second, moving up and to the right of the origin, the level of utility increases.
2. Assumption (49) implies that any given indifference curve is convex towards the origin.
In turn, this implies that, along any given indifference curve, it is increasingly difficult
for the agent to substitute future consumption with current consumption.
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Figure 12: Graphical representation of the indifference curves in C and C’
Marginal rate of substitution. The marginal rate of substitution between C and
′
C is given by the ratio of the marginal utilities:
M RSC,C ′ ≡
UC (C)
∂U/∂C
′ ≡
∂U/∂C
UC ′ (C ′ )
In particular, −M RSC,C ′ measures the slope of any given indifference curve in the point
′
(C, C ). Given the assumption of convexity, the MRS is decreasing (in absolute value) along
any given indifference curve.
Optimality condition for the household. The optimality condition of the household’s problem reads:
UC
= 1| {z
+ r}
UC ′
| {z } (−)slope of
PV budget
constraint
(−)slope of
indiff.
curves
The above equation states that, if the consumer is at the optimum,22 the (negative of the)
slope of the intertemporal budget line must be equal to the (negative of the) slope of the
highest indifference curve.
22
Recall that this is only a necessary condition for an optimum.
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Figure 13: Optimal consumption choice is at point A. In the current period the consumer
acts as a lender.
′
Graphically, the optimal consumption choice, labeled (C∗ , C ∗ ) for a lender can be
represented in Figure 13:
Intuition for optimality condition. We can rewrite the consumption optimality
condition in a more intuitive way as follows:
UC
|{z}
marginal
utility of C
in period 1
= UC ′ (1 + r)
| {z }
(53)
marginal utility
of saving
This expression states that, at the optimum, the consumer must be equating the marginal
utility of consumption (the left hand side) to the marginal utility of saving (right hand side).
This equation is typically called the consumption Euler equation. What is the intuition for
this condition? The agent’s problem is the one of choosing, for any given additional unit of
endowment (“manna”), whether to allocate that unit to (i) consumption; or (ii) savings. Plan
(i) yields utility UC (i.e., the utility of a marginal/additional unit of manna from heaven);
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plan (ii) (i.e., saving) allows to postpone the consumption of the additional unit of manna.
That additional unit can be saved (via the credit market) and transferred into the future,
where it becomes (1 + r). Notice that 1 + r is the return from saving one marginal unit of
manna (it is therefore expressed in “units of manna”). Once expressed in units of utility,
1+r units of manna become UC ′ (1+r). What condition (53) therefore states is that in units
of utility a marginal unit of manna devoted to consumption must be equated to a marginal
unit devoted to saving. In fact, if the left-hand-side of (53) were greater (smaller) than the
right-hand-side, then the consumer could increase her intertemporal utility by consuming
more (less) in the current period. At the optimum the consumer must be indifferent (in
terms of utility) between consuming an extra unit of manna in period 1 and consuming 1+r
extra units in period 2.
Mathematical derivation of the efficiency condition. The constrained maximization problem in (50) can be expressed as a less constrained problem by combining the
period by period budget constraints (51) and (52) into a single constraint. We already know
that this yields the present value budget constraint (44). Let λ be the Lagrange multiplier
on constraint (44). We can therefore write Lagrangean problem
′
′
C
Y
L≡ U (C, C ) − λ C +
−Y −
1+r
1+r
′
′
The first order conditions with respect to C and C are:
UC − λ = 0
UC ′ −
(54)
λ
=0
1+r
(55)
In addition, optimality requires:
′ ′
Y
C
=0
−Y −
λ· C +
1+r
1+r
Hence the present value budget constraint holding with equality requires
λ > 0→
′
′
C
Y
C+
= Y +
1+r
1+r
Substituting λ > 0 into (55) one obtains (53).
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9.5
Experiments with the optimal consumption model
1. Increase in current income
2. Increase in future income
3. Effect of a change in the real interest rate
4. Temporary vs. permanent changes in income
9.5.1
Consumption smoothing
An important question to analyze is the following: what happens to consumption if current
income increases (holding constant future income)? This corresponds for example to winning
a lottery. Does consumption increase in a 1:1 proportion? Notice first that this leads to an
increase in lifetime wealth. Hence the PV budget constraint shifts to the right (the slope is
unchanged as real interest rate r is constant):
′ Y
↑ we ≡ ↑ Y +
1+r
Hence both current and future consumption increase. In Figure 14 the optimal consumption point shifts from point A to point B. Intuitively the consumer wishes (optimally) to
spread the increase in consumption between today and tomorrow. This principle is known
as consumption smoothing. Graphically this can be seen by noticing that the increase in
income is measured by AD, whereas the increase in current consumption is measured by
AF . Hence part of the increase in current income is saved. Importantly this is the result
of an optimal, forward-looking, choice by the consumer given her preferences. In particular,
′
this result is an implication of (i) both C and C being normal goods; (ii) the convexity of
the indifference curves.
9.5.2
Analytical example
Suppose the utility function is given by:
U (·) = log C + β log C
The optimality condition reads:
′
β ∈ [0, 1]
1
β
= (1 + r) ′
C
C
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Figure 14: The effect of an increase in current income on current and future consumption.
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′
Combine with the PV budget constraint (assume once again for simplicity that T = T = 0)
′
′
C+
C
Y
=Y +
1+r
1+r
(57)
For any given real interest rate, the system (56), (57) has 2 equations in 2 unknowns
′
(C, C ). From (56):
′
C = (1 + r)βC
(58)
Substituting into (57):
(1 + r)βC
C+
1+r
′
Y
= Y +
1+r
′
Y
C(1 + β) = Y +
1+r
Finally we obtain:
′ 1
Y
C=
Y +
≡ C∗
(1 + β)
1+r
(59)
For a given real interest rate, equation (59) expresses current consumption as a function
of the present value of income. The latter is usually defined as permanent income. Our
experiment:
1
∂C
=
<1
∂Y
(1 + β)
→ consumption smoothing
Hence from the viewpoint of the individual consumer (who takes the real interest rate as
given), consumption responds less than one for one to current income.
′
The solution for future consumption C can be derived by substituting (59) into (58):
C
′
′ Y
β(1 + r)
Y +
=
(1 + β)
1+r
≡ C
(60)
′∗
Notice that future consumption increases in response to a temporary increase in income:
′
∂C
β(1 + r)
=
>0
∂Y
(1 + β)
This result is consistent with our assumption of both current and future consumption being
normal goods.
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When is an agent a saver (borrower)? What determines whether, at the optimum,
the agent chooses to be a saver (as opposed to a borrower)? For an agent to choose to be
a saver it must hold that, at the optimum, current consumption does not exceed current
income:
C = C∗ ≤ Y
As a result of an optimal choice, current consumption is given by (59). Therefore:
Rearranging one obtains:
′ 1
Y
Y +
≤Y
(1 + β)
1+r
′
1
Y
β
≤Y
(1 + β) 1 + r
(1 + β)
which finally yields:
′
Y
≤ (1 + r)β
Y
′
Intuitively, the agent will choose to be a saver (borrower) if the growth rate of income Y /Y
is sufficiently low (high).
9.6
Permanent income hypothesis
Is winning a lottery different from a rise in the paycheck? Does this produce different effects
on consumption? According to Friedman (1957), individual consumption behaves according
to the permanent income hypothesis (PIH): consumption is tightly linked to household’s
permanent income, i.e., to lifetime wealth in our model. Notice that this is very different
from the consumption function assumed in the IS-LM model, where current consumption
depends on current disposable income only, via some function C(·):
C = C(Y )
Typically C(·) is assumed to be linear:
C = c0 + c1 (Y )
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Does our model satisfy the PIH? Consider again our example. The solution for consumption
is:
′ 1
Y
C=
Y +
(1 + β)
1+r
(61)
1. Increase in current income only:
∂C
1
=
∂Y
(1 + β)
2. Increase in permanent income. By totally differentiating (61):
dC =
1
1
′
dY +
dY
(1 + β)
(1 + β) (1 + r)
′
Since dY = dY →
dC
1
=
dY
(1 + β)
2+r
1+r
>
1
(1 + β)
Hence for a given real interest rate, an increase in permanent income generates a larger effect
on current consumption relative to an increase in current income.
10
Competitive equilibrium and endogenous real interest rate
In a competitive equilibrium for this two-period economy, the following conditions must hold:
1. The consumer chooses first- and second-period consumption and savings optimally
given the real interest rate r.
2. The period-by-period budget constraints of the household are satisfied (and, as an
implication, also the present value budget constraint).
3. The credit market and the goods market both clear.
Market clearing in the credit market requires:
S=0
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Implications for market clearing. We now want to show that if the credit market
equilibrium holds and budget constraints are satisfied with equality, then all resources in
the aggregate economy are exhausted. This is the so called aggregate feasibility constraint,
or, alternatively, the market clearing condition in the goods market, stating that, at the
equilibrium, the aggregate supply of goods must be equal to the aggregate consumption of
goods.
Note first that
S =Y −C
(from the private budget constraint)
(63)
Using equilibrium condition (62) we have:
Y −C =0
(64)
=
(65)
Therefore:
Y
|{z}
C
|{z}
total
expenditure
production
Note that equation (65) was typically assumed as an identity in the IS-LM model. Here it’s
been derived as an implication of the equilibrium. The condition above, however, is not an
independent equilibrium condition, due to Walras’ law.
We can therefore state the following formal definition of a competitive equilibrium.
′
For any given sequence of endowments Y, Y , a competitive equilibrium in the two-period
′
economy without a government is a set of allocations for C, C , S , r solving the following
set of equations:
C =Y −S
(66)
C = Y + (1 + r)S
(67)
S=0
(68)
′
UC
=1+r
(69)
UC ′
The one above is a system of four equations in four unknowns, and can therefore be solved.
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The real interest rate. Notice that in the system (66)-(69) the real interest rate
is now treated as an endogenous variable. In fact, the real interest rate is precisely the
price ensuring that the credit market is in equilibrium, i.e., that S = 0 always holds. The
intuition is as follows. Suppose for instance that income in the current period, Y , increases.
For given consumption, this would tend to increase savings. But since S = 0 must hold in
equilibrium, the real interest rate must adjust exactly to offset the increased desire to save,
i.e., the real rate must decrease. This is precisely the movement in the interest rate that
stimulates consumption, so that the increase in income is matched (in equilibrium) by the
increase in consumption (and Y = C is satisfied).
Consumption smoothing vs. market clearing. As derived above, it is a feature
of this equilibrium model that consumption, unlike the standard IS-LM model, is a function
of permanent income, and not of current income. That is an implication of consumption
smoothing, which in turn depends on the concavity of the utility function. Suppose for the
sake of illustration that the utility function is once again logarithmic and separable. The
expression for consumption is:
1
C=
1+β
′
Y
Y +
1+r
(70)
1
Therefore current consumption is a constant fraction 1+β
of the present value of income.
′
Consider, at a given interest rate, a rise in future income Y : this generates a rise in current
consumption equal to
1
∂C
∈ (0, 1)
′ =
∂Y
(1 + β)(1 + r)
Hence current consumption does react to future income and with a sensitivity smaller
than 1. This embeds the idea of consumption smoothing, whereby the agent starts consuming
more today despite the fact higher income will materialize only in the future. The important
point is however the following: how does consumption smoothing square with the goods
market equilibrium condition that indicates that C = Y , i.e., that current consumption is
equal to current income in every period? The answer is that the reasoning above holds at
a given real interest rate. Therefore the smoothing of consumption is a behavior that the
agent would like to adopt from her own individual perspective, i.e., taking the real interest
rate (which is the market price of current consumption relative to future consumption, or
the opportunity cost of consuming today as opposed to tomorrow) as given. In equilibrium,
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′
however, the real interest rate will rise in response to a rise in Y . The rise in the real
interest rate is precisely the equilibrium movement that in equation (70) insures that current
consumption remains constant and C = Y is satisfied.
In order to see clearly the behavior of the real interest rate in the equilibrium, let’s
substitute the condition S = 0 into the period by period budget constraints (66) and (67),
which yields:
′
C =Y; C =Y
′
(71)
Condition (69) under the assumption of log-utility once again reads:
1
(1 + r)
=β
(72)
C
C′
Substituting (71) we obtain the equilibrium solution for the real interest rate (i.e., an expression of the real interest rate only as a function of exogenous variables and parameters):
′
(1 + r) =
Y
Yβ
(73)
Hence we see that, in equilibrium, the real interest rate must rise in response to a rise in
future income:
∂(1 + r)
1
=
>0
′
∂Y
Yβ
General equilibrium reasoning. The general equilibrium effect of a rise in future
′
income Y can be described as follows. At a given real interest rate r, a rise in future income
(due to consumption smoothing) would lead to a rise in current consumption. From a logical (and mathematical) viewpoint this seems to contradict the market clearing (equilibrium)
condition that states that consumption is equal to income in every period, C = Y : since
current income Y has not changed by assumption, how can current consumption rise? The
answer is in the behavior of the real interest rate. Since the condition C = Y , or equivalently S = 0, must be satisfied (otherwise our definition of equilibrium would be violated)
the real interest rate will have to rise (see equation (73)) precisely to discourage current con′
sumption. Formally, on the right hand side of equation (70), the rise in Y will be exactly
compensated by a rise in r in order to keep current consumption C constant. Put differently,
at a given initial real interest rate, the rise in future income would induce the agent (today)
to consume more than its current income, i.e., it would induce the agent to borrow. But at
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the equilibrium we know that net borrowing (or saving) must be zero. Hence something will
have to happen to the relative price of saving in order to discourage borrowing (encourage
saving), and that is precisely a rise in the real interest rate.
To summarize, at the equilibrium, a rise in future income will produce two effects: (i) a
rise in future consumption; and (ii) a rise in the real interest rate, with current consumption
remaining constant.
11
Introducing the government
′
The government finances an exogenous stream of government spending G, G
either via
(i) taxes or (ii) debt (B > 0), or both. Its budget constraint in period 1 is given by:
(74)
G = T + |{z}
B
govt. debt
The government budget constraint in period 2 reads:
′
G + (1 + r)B = T
| {z }
′
(75)
repay interest
on debt
Notice that the government faces the same real interest rate r faced by the households.
This implicitly assumes that there do not exist financial market imperfections that might
eventually drive a wedge between the real interest rate faced by the households and the one
faced by the government.
Balanced budget. Notice that if the government were to balance the budget in every
period we would have:
′
G = T; G = T
′
Government present-value budget constraint. In a way similar to the household
case we can derive an intertemporal budget constraint also for the government. Substituting
B from (75) we obtain:
′
T −G
B=
1+r
and therefore
′
′
′
G
T
G+
=T+
1+r
1+r
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Hence, for the government, the present value of government spending cannot exceed the
present value of tax revenues (i.e., of government’s income). The terminal condition for the
government will read:
′
B =0
The household’s intertemporal budget constraint reads:
′
′
′
Y
T
C
=Y +
−T −
C+
1+r
1+r
1+r
(77)
Substituting (76) into (77) one obtains:
′
′
′
C
Y
G
C+
=Y +
−G−
1+r
1+r
1+r
(78)
Competitive equilibrium with a government. In a competitive equilibrium for
the two-period economy, three conditions must hold:
1. Each consumer chooses first- and second-period consumption and savings optimally
given the real interest rate r.
2. The private agent present value budget constraint and the government present-value
budget constraint holds.
3. The credit market and the goods market both clear
Market clearing in the credit market requires:
S = |{z}
B
|{z}
private
saving
(79)
govt.
borrowing
Once again we wish to show that if the credit market equilibrium holds and budget constraints are satisfied with equality, then all resources in the aggregate economy are exhausted
(aggregate feasibility). Note first that
S =Y −C −T
B =G−T
(from private budget constraint)
(80)
(from govt. budget constraint)
(81)
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Using (79) we have
Y −C −T =G−T
(82)
Therefore rearranging:
Y
|{z}
production
(83)
= |C {z
+ G}
total
expenditure
We can therefore state the following definition of a competitive equilibrium with a govern
′
′
ment. For any given sequence of endowments Y, Y and of government spending G, G
′
a competitive equilibrium is a set of allocations C, C , r solving the following set of
equations:
′
′
′
C
Y
G
C+
=Y +
−G−
1+r
1+r
1+r
(84)
Y =C +G
(85)
UC
=1+r
UC ′
(86)
′
The one above is a system of 3 equations in 3 unknowns C, C , r , and can therefore be
′
solved. Notice that in (84)-(86) the specification of the path of taxes T, T and the level
of government debt B are both irrelevant for the determination of the equilibrium. This
insight already captures the main intuition for our next topic, i.e., Ricardian Equivalence.
11.1
Ricardian Equivalence theorem
Next we introduce the so-called Ricardian Equivalence theorem. This theoretical result is
a useful benchmark to analyze the macroeconomic effects of alternative ways of financing a
spending policy by the government. The Ricardian Equivalence theorem is often presented
as a result of “neutrality” of fiscal policy. For instance: for a given stream of government
spending (present and future), and holding constant the real interest rate, a variation in
(lump-sum) taxes does not affect the level of private consumption. While this is true in
a simple intertemporal model with lump-sum taxes and perfect credit market, it is only
a corollary of Ricardian Equivalence. The correct, and more general, formulation of the
Ricardian Equivalence theorem is different.
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Definition of Ricardian Equivalence. For any given stream of (current and future)
government spending, it is irrelevant for the determination of equilibrium consumption and
real interest rate whether that spending is financed either via (i) taxes or (ii) government
debt.
It is important to notice that Ricardian Equivalence is a theorem of irrelevance of the
source of financing of government spending, and not a result of neutrality of fiscal policy on
consumption and/or production.
11.1.1
Equivalence between balanced-budget and government debt
Consider the usual two-period endowment economy (i.e., output is “manna from heaven” in
both periods). The intertemporal budget constraint of the private household is:
′
′
C
Y −T
C+
=Y −T +
1+r
1+r
′
(87)
′
The government must finance a given stream of government spending (G, G ). We
consider two options of financing:
′
1. Taxes in both periods: T, T ;
2. Issuing government debt B in the first period and repay in the future period.
Case 1: only taxes (balanced budget policy). Suppose the government issues no
debt:
B=0
Therefore it decides to finance government spending each period via a balanced budget:
G=T
′
G =T
(88)
′
(89)
Substituting (88) and (89):
′
′
C
Y −G
C+
=Y −G+
1+r
1+r
′
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Case 2: government debt policy. Suppose alternatively that the government finances government spending only by issuing debt in the current period. Therefore:
T =0
(91)
′
Notice that we cannot assume T = 0, since future government spending (including the
service cost of debt) will have to be financed. The intertemporal budget constraint of the
government reads in this case:
′
′
G
T
G+
= T+
1+r
1+r
′
T
=
1+r
(92)
Substituting (91) and (92) into (87) one can rewrite the intertemporal budget constraint of
the private sector:
′
′
′
Y −T
C
= Y −T +
C+
1+r
1+r
′
′
Y
T
= Y −0+
−
1+r 1+r
′
′
Y
G
= Y +
−G−
1+r
1+r
(93)
Notice that the last expression in (93) is identical to (90). The message is very simple,
but very general. From the point of view of the household, the following two options are
equivalent: (i) the government runs a balanced budget in every period; (ii) the government
issues debt and repays the same debt tomorrow by raising future taxes. In both cases, the
intertemporal budget constraint of the household is unaffected, and therefore her consumption choices (at a given real interest rate). Equivalently, and as implied directly from (93),
we can state that all that matters for consumption (current and future) is simply the current and future values of government spending: in fact, neither taxes (current or future) or
government debt are featured on the right hand side of (93).
Let’s evaluate the equivalence between a balanced-budget policy and a government debt
policy in more detail, by looking at the period-by-period constraints.
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1. Balanced-budget policy:
G=T
′
G = T′
From the private budget constraint
C =Y −T −S
′
′
′
C = Y − T + (1 + r)S
(94)
(95)
where S is private savings. In equilibrium (since there is no issuance of government
debt):
S=B=0
This implies
C = Y −T
(96)
= Y −G
C
′
= Y −T
′
′
′
′
= Y −G
(97)
2. Government debt:
In this case we have
G=B
′
G + (1 + r)B = T ′
The household’s period by period budget constraints:
C = Y −T −S
= Y −S
= Y −B
= Y −G
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C
′
′
′
′
′
= Y − T + (1 + r)S
h ′
i
′
= Y − G + (1 + r)B +(1 + r)S
{z
}
|
(99)
T′
= Y −G
Once again we see that the period-by-period budget constraints of the households are
identical in the two scenarios, i.e., balanced-budget policy vs. government debt.
11.1.2
Corollaries of Ricardian Equivalence
Sometimes the Ricardian Equivalence theorem is presented as a proposition of neutrality of
fiscal policy (on consumption and/or output). This statement, however, is only a corollary
of Ricardian Equivalence and holds, under certain circumstances, for tax policy only. In fact
we can state that the following is true: if fiscal policy (tax policy in particular) is neutral,
then Ricardian Equivalence holds. But the reverse is not generally true. Therefore we can
state:
Fiscal policy neutral
→
Ricardian Equivalence holds
↚
The idea behind the above proposition is, very simply, that if fiscal policy is conducted
via variations in government spending, those changes are not neutral on consumption, but
nevertheless Ricardian Equivalence holds. Let’s consider taxes and government spending
separately.
Tax policy. Consider a change in current taxes ∆T , holding constant the path of
′
government spending (G, G ). This means:
′
∆G = ∆G = 0
From the intertemporal budget constraint of the government therefore it must hold:
′
∆T
∆T = −
1+r
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But this also means that the intertemporal budget constraint of the household will be
unaffected, because the present value of net income is unaffected. The optimal consumption
choice is therefore unaffected by the tax policy.
Government spending policy. Consider now a change in government spending ∆G.
We know that in equilibrium it must be true that
Y
= C +G
Y
= C +G
′
′
Since Y is given, it must be the case that current consumption falls. Therefore fiscal policy
looses its property of neutrality. Nevertheless, Ricardian Equivalence still holds. The logic
follows from equation (93), which we rewrite here:
′
′
′
C
Y
G
C+
=Y +
−G−
1+r
1+r
1+r
(101)
Recall that the above equation follows from combining the intertemporal budget constraints of both the private sector and the government. When that is done, all that matters
′
for consumption is the path of government spending G, G , as the right hand side of (101)
suggests, regardless of the level of government debt and/or taxes. But this is exactly the
Ricardian Equivalence logic: whether this stream of government spending is financed via
a balanced-budget policy as opposed to government debt, is completely irrelevant. What
matters is the size of the spending package, not the composition of its financing.
Ricardian equivalence in the competitive equilibrium with government. It
is straightforward to notice that Ricardian Equivalence holds in the formulation of the competitive equilibrium with a government. It is sufficient to notice that, in the equilibrium,
′
the triple of endogenous variables C, C , r can be determined independently of the level
of taxes and/or government debt in each period. In other words, all that matters for the
′
determination of the equilibrium is the sequence of government spending G, G , irrespective of its form of financing. A corollary of this argument is that the timing of taxes (i.e., T
′
vs T ) is irrelevant.
A more formal way to state the above result is that the composition of financing
B, T, T
′
is indeterminate - and therefore irrelevant for the equilibrium. In fact, given
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n
′
′
C = C, C = C , r = r
o
determined from the system (84)-(86) above, the remaining set
′
of equilibrium conditions is overdetermined to pin down the triple B, T, T , i.e., the num-
ber of equations exceed the number of endogenous variables:
G=T +B
′
G + (1 + r)B = T
(102)
′
(103)
C = Y − T − B=S
′
′
(104)
′
C = Y − T + (1 + r)B
11.1.3
(105)
A tax experiment
Starting from the equilibrium, consider the following tax experiment. The setup is as usual:
(i) the consumer maximizes utility; (ii) equation (87) holds (PV budget constraints holds);
(iii) the credit market equilibrium holds:
S =B → Y =C +G
In this setting, consider a change in current taxes under two conditions:
1. Holding constant the PV of government spending
′
′
G
T
G+
=T+
1 + r}
1+r
| {z
(106)
constant
2. Unchanged real interest rate r
It follows that for the PV budget constraint of the government to continue to hold, the
following condition must be true:
∆T +
1
′
∆T = 0
1+r
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Rearranging the above equation we obtain:
∆T = −
1
′
∆T
1+r
(108)
Hence the experiment corresponds to a “change in the timing of taxes.” But since this leaves
the PV budget constraint of the consumer unaltered, the optimal consumption choice does
not change. Why?
Suppose a cut in taxes today: ∆T < 0. Since the government keeps the PV of government spending unaltered, it needs to finance the current lack of resources (∆T < 0)
by increasing government debt. But new government debt must be repaid in the future,
′
therefore future taxes need to be increased: ∆T > 0
1
′
∆T = −
∆T
|{z}
1+r
<0
{z
}
|
<0
Let’s see this in detail from the period-by-period budget constraints of the government:
∆G = |{z}
∆T + |{z}
∆B
|{z}
=0
This implies
<0
(109)
>0
∆T = −∆B
On the other hand, in period 2:
′
∆G + (1 + r)∆B = ∆T
′
(110)
It follows that taxes need to be increased by:
∆T
′
= (1 + r)∆B
(111)
= −(1 + r)∆T
which is exactly equation (108) derived above.
11.1.4
Effects on private and public saving
Consider once again the cut in current taxes analyzed above (∆T < 0). If the government
increases borrowing as a result of the tax cut, it must be the case that households lend
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to the government. How much do the households save today? Precisely the amount they
anticipate they will have to repay in the following period when future taxes will be increased
′
by ∆T = −(1 + r)∆T . To see this, note that the increase in private savings today is:
∆S = ∆Y − ∆T − ∆C
= 0 + ∆B + 0 > 0
private saving
On the other hand, the change in government saving (or borrowing):
∆S g = ∆T − ∆G
= −∆B < 0
government saving
Credit market equilibrium under Ricardian Equivalence. Note that the effects
on savings described above must be such that the “price” of saving - the real interest rate r
- remains constant. Figure 15 illustrates this case.
11.2
Implications of Ricardian Equivalence
The Ricardian Equivalence result is extreme, but with profound meaning. We can think of
at least two main implications.
1. A tax cut is not a “free lunch.” This means that a cut in current taxes must be
financed by an increase in government debt, but the latter must be repaid and financed
with an increase in future taxes. Therefore government debt shifts the burden of
taxation simply into the future. However, any forward-looking agent anticipating this
will perceive her intertemporal budget constraint as unchanged, and therefore will not
alter her consumption choice.
2. Intertemporal budget constraints are important. This corollary is pervasive in the whole
analysis so far. The fact that the relevant budget constraints are genuinely intertemporal follows from one of the main pillars of our setting, i.e., the fact that we are
considering a dynamic (intertemporal) economy.
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Figure 15: Effect a cut in lump sum taxes on the credit market equilibrium under Ricardian
Equivalence. Note: in the figure S p = S denotes private savings, B denotes government
debt.
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11.2.1
Assumptions under which Ricardian Equivalence holds
It is important to clarify that Ricardian Equivalence holds under four precise assumptions.
1. When taxes change they change by the same amount for all consumers, both in the
present and in the future.
2. Any debt issued by the government is paid off during the lifetime of the people alive
when the debt was issued.
3. All taxes are lump-sum.
4. Credit markets are perfect: consumers can borrow/lend freely subject to their present
value budget constraints, and they can borrow and lend at the same interest rate as
the government.
11.2.2
Variations in government spending
We can now analyze the effects of variations in government spending, keeping into account
the equilibrium response of the real interest rate. Consider a rise in current G financed by
an increase in taxes:
∆G = ∆T
From the goods market equilibrium condition we know:
Y =C +G
(112)
Recall that the above condition is in turn implied by the equilibrium condition on the credit
market. The equilibrium in the credit market requires that aggregate savings (including both
private and public) be equal to zero. Private savings read:
S ≡Y −T −C
Public savings are:
Sg = T − G
The equilibrium condition therefore is:
S + Sg = 0
(Y − T − C) + (T − G) = 0
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which in turn leads to (112).
Notice that public savings S g are constant, since the increase in G is matched by an
increase in current taxes T . The increase in taxes, however, leads to a fall in private savings
S for any given level of consumption. Let’s define ∆S0 < 0 the (hypothetical) fall in savings
due to the mere rise in taxes, holding consumption constant. For condition (113) to hold,
however, private savings must remain unchanged. The rise in the real interest rate insures
precisely an upward variation in savings, say ∆S1 > 0, such that ∆S0 + ∆S1 = 0, and so
that equation (113) is satisfied.
Analytically, consider once again the case of log-separable utility, which leads to the
first order condition (69):
1
β
= ′ (1 + r)
(114)
C
C
Substituting C from (112), we obtain the following expression for the equilibrium real interest
rate:
(1 + r) =
′
Y −G
Y −G
′
1
β
(115)
Notice that (115) is the model solution (reduced form) for the real interest rate, since only
exogenous variables and parameters are featured on the right hand side. Hence a rise in
current G leads to a rise in the equilibrium real interest rate:
∂(1 + r)
>0
∂G
A similar reasoning holds in the case in which the increase in G is financed by government
debt. In fact, the expression for the equilibrium real interest rate is identical to (115). This,
once again, is an implication of Ricardian equivalence. The composition of private and
government savings, however, will differ relative to the balanced-budget case. Since taxes
are constant, government savings S g ≡ T −G will fall. Private savings S, therefore, will have
to rise. This will be induced precisely by a rise in the real interest rate, which discourages
consumption.
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12
Production and real business cycles
In this section introduce production in the economy.23 We assume that output is no longer
“manna from heaven,” but it is produced in period 2 by firms which use physical capital
generated via investment in period 1. The following setup aims at illustrating, in a simplified manner, the pillars of the Real Business Cycle theory initiated by F. Kydland and
R. Prescott, which took stock of the Lucas revolution. With RBC theory the emphasis
shifts from (unanticipated) monetary shocks towards a role of real (productivity) shocks as
fundamental drivers of aggregate fluctuations. What remains at the heart of the analysis,
however, is the methodological role of general equilibrium, microfoundations and rational
expectations.
12.1
Firms
We assume the existence of a representative firm which produces output in period 2 using
capital holdings K at the beginning of period 2 according to the production function
Y2 = A2 F (K1 )
(116)
where A2 is an exogenous productivity factor.24 The function F (·) is increasing and concave:
FK (·) > 0
(117)
FKK (·) ≤ 0
(118)
The latter assumption implies that there are diminishing returns to capital.
We assume that the economy is born in period 1 with an exogenous capital stock K0
previously accumulated. Let investment in physical capital in period 1 be denoted by I1 .
The capital stock depreciates across periods at the rate δ. The capital accumulation equation
therefore reads:
23
For an extended reference see Schmitt-Grohe, Uribe and Woodford (2016), available at
http://www.columbia.edu/˜mu2166/UIM/index.html
24
RBC models typically specify a role for endogenous labor supply. Lucas and Rapping (1969) is a seminal
contribution studying the foundations of intertemporal labor supply decisions in an equilibrium setup. We
abstract from the role of labor here.
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K1
|{z}
= (1 − δ)K0 +
I1
|{z}
(119)
investment
flow in
period 1
capital stock
at the beginning of
period 2 (= end of
period 1)
To simplify our analysis we will assume that physical capital fully depreciates between the
two period, i.e., δ = 1. From (119) we therefore can write:
K1 = I 1
The production function in period 2 is rewritten
Y2 = A2 F (I1 )
(120)
Output in period 1 is exogenously given.25 Therefore in this model only period 2 output is
endogenous: it depends on the firm’s investment decision taken in period 1.
Firm’s problem. In period 1 the firm borrows from households26 in order to finance
the purchase of physical capital. Let D1f denote firm’s borrowing in period 1 (expressed in
real units). Hence we have:
I1 = D1f
The cost of borrowing is given by the (net) real interest rate r1 . Hence period 2 profits read:
Γ2 = A2 F (I1 ) − (1 + r1 )D1f
(121)
= A2 F (I1 ) − (1 + r1 )I1
= Γ(I1 )
The first order optimality condition is obtained by differentiating Γ(·) with respect to I1 :
25
More specifically output in period 1 would be given by
Y1 = A1 F (K0 )
Since A1 is exogenous and K0 is exogenous it follows that also Y1 is exogenous.
This setting will change in the open economy, where it will be assumed that firms borrow in international
financial markets.
26
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′
A2 F (I1 ) − (1 + r1 ) = 0
Rewriting:
′
A2 F (I1 ) = (1 + r1 )
(122)
The above condition states that, at the optimum, the firm will equate the marginal product of
′
capital, A2 F (I1 ), to the marginal cost of investment (the latter being given by the financial
cost of borrowing).
Investment schedule. Equation (122) allows to derive an equilibrium investment
schedule, whereby investment is expressed as a function of the real interest rate for any
given level of productivity. Consider the effect of a higher real interest rate r on investment.
Since A2 is given, I1 will have to change in order for the right hand side of equation (122) to
′
rise. Since F (I1 ) is decreasing in I1 (due to the concavity assumption (118)), I1 will have
′
to fall in order for F (I1 ) to rise. We therefore conclude that, at the equilibrium (i.e., taking
into account firm’s optimal investment decision), and for any given value of productivity A2 ,
investment is a decreasing function of the real interest rate.
(−)
I1 = I(r1 , A2 )
In turn, holding constant the real interest rate, investment is an increasing function of
productivity. Since r1 is constant on the right hand side of (122), a rise in A2 will require
′
a rise in I1 (and therefore a fall in F (I1 ), due to the concavity assumption (118)) for the
left hand side to remain constant. Figure 16 depicts the production function, which is an
increasing function of investment for any given value of period 2 productivity A2 . The
′
production function features diminishing marginal returns, i.e., the slope A2 F (I1 ), which
is the marginal product of capital, decreases with investment for any given value of A2 . In
light of an exogenous rise in productivity to A+
2 > A2 , and for any given value of investment
I1 , the production function shifts upward, thereby raising period 2 output Y2 .
Productivity and investment: partial equilibrium. Figure 17 depicts the firm’s
optimality condition, equating the marginal benefit of additional investment (i.e., the marginal
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Figure 16: Production function: effect of a rise in productivity
product of capital) to the marginal cost (i.e., the real interest rate). Consider a rise in productivity from A2 to A+
2 . For a given level of the real interest rate, for equation (122) to
′
continue to hold, investment must rise, in order for the term A2 F (I1 ) to remain constant
′
(recall that F (I1 ) is a decreasing function of investment). Optimal firm’s investment raises
from I1∗ to I1∗+ . It is important to notice, though, that this result is derived under a partial
equilibrium assumption, i.e., holding the real interest rate constant.
12.2
Households
The representative household is the owner of the firm, and is therefore entitled to firm’s
profits. Her budget constraint in period 1 reads:
C1 +
S1h = Γ1
|{z}
(123)
household
saving
where S1h is holdings of bonds at the end of period 1. Notice that profits in period 1 are
exogenous. Hence total household’s income in period 1, Γ1 , is exogenous from the viewpoint
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Figure 17: Marginal product schedule: effect of a rise in productivity on the optimal level
of investment for a given level of the real interest rate (partial equilibrium).
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of the household.
The budget constraint in period 2 reads:
C2 + S2h = Γ2 + (1 + r1 )S1h
Household’s financial wealth at the end of period 2 must be equal to zero:
S2h = 0
which implies
C2 = Γ2 + (1 + r1 )S1h
(124)
Combining (123) and (124) to eliminate S1h we obtain the household’s intertemporal budget
constraint
C1 +
Γ2
C2
= Γ1 +
(1 + r1 )
1 + r1
The present value of consumption is equal to the present value of income (given by the
present value of profits), which in turn is given by the present value of profits. Notice that,
unlike our previous endowment economy, the component Γ2 of the present value of income
is endogenous.
As in our baseline setting, the optimal consumption choice of the agent, taking the real
interest rate as given, reads:
UC1 (C1 )
= 1 + r1
UC2 (C2 )
12.3
Equilibrium in the production economy
In a competitive equilibrium, the credit market must clear. This implies that savings by the
household must match debt holdings by the firms:
S1h = D1f
Using the definition of firm’s profits, we can rewrite the household’s budget constraints:
C1 + S1h = A1 F (I0 )
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C2 = A2 F (I1 ) − (1 + r1 )D1f + (1 + r1 )S1h
An
n equilibrium in the oeconomy with production is an allocation for six endogenous variables
C1 , C2 , I1 , S1h , D1f , r solving the following system of six equations:
UC1 (C1 )
= 1 + r1
UC2 (C2 )
′
(126)
A2 F (I1 ) = 1 + r1
(127)
C1 + S1h = A1 F (I0 )
(128)
C2 = A2 F (I1 ) − (1 + r1 )D1f + (1 + r1 )S1h
(129)
S1h = D1f
(130)
I1 = D1f
(131)
for given values of {A1 , A2 , I0 }.
It is immediate to rewrite the above system of equilibrium conditions in a more compact
form. Substituting (130) and (131) yields:
UC1 (C1 )
= 1 + r1
UC2 (C2 )
′
(132)
A2 F (I1 ) = 1 + r1
(133)
C1 + I1 = A1 F (I0 )
(134)
C2 = A2 F (I1 )
(135)
The latter is a system of four equations in four unknowns: {C1 , C2 , I1 , r1 }. Equation
(132) is the household’s consumption optimality condition. Equation (133) is the firm’s
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optimality condition (equating the marginal product of capital to the real interest rate).
Finally, equations (134) and (135) are the good market clearing conditions in period 1 and
period 2 respectively. Notice that the equilibrium in the goods market in period 1 now
includes investment on the demand side, whereas output is exogenous on the supply side. In
period 2, equilibrium in the goods market features endogenous output on the supply side,
whereas the demand side is only given by consumption.
12.3.1
Productivity and investment in general equilibrium
Given the system of equilibrium conditions (132)-(135), we can now try to derive the general
equilibrium implications of variations in (current and future) productivity. This will be easily
done by assuming the following functional forms for utility and production function:
U (C1 , C2 ) = log C1 + β log C2
(136)
F (Kt ) = Ktα , α < 1 (t = 0, 1)
(137)
From (132) and (133) we can write:
C2
= β(1 + r1 )
C1
= βαA2 I1α−1
(138)
Since C2 = A2 I1α (from 135), we have:
A2 I1α
= βαA2 I1α−1
C1
which allows us to solve for the equilibrium investment-consumption ratio:
I1
= αβ
(139)
C1
Hence the investment-consumption ratio is constant in equilibrium and independent of the
level of productivity.
Substituting (139) into (134) we can solve for consumption in period 1:
C1 =
1
α
A1 I 0
|
{z
}
1 + αβ
Y1
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where Y 1 indicates that output in period 1 is exogenous.
We can then solve for investment in period 1 from (134):
1
Y 1 +I1 = Y 1
1 + αβ
| {z }
(141)
C1
which yields:
I1 =
αβ
α
A1 I
1 + αβ | {z 0}
(142)
Y1
Hence investment in period 1 does not depend on productivity in period 2. This result
seems to contradict the result depicted in Figure 17. The key however lies in the general
equilibrium response of the real interest rate, which is kept constant in (17). In response to
a rise in future productivity A2 , not only the marginal product of capital will rise (making
investment more attractive); the real interest rate will also rise, making investment financially
more costly. In turns out that given the functional forms specified in (136) and (137) these
two forces exactly balance each other out, with investment remaining constant in equilibrium
in response to a rise in future productivity. As equation (142) shows, however, current
αβ
investment responds positively to current productivity, with elasticity 1+αβ
.
Using (135) and (142), we can then derive an expression for future consumption C2 :
C2 = A2 I1α
α
= A2 Y 1
αβ
1 + αβ
(143)
α
In order to gauge the general equilibrium response of the real interest rate, let’s substitute (140) and (143) into (138), and obtain:
β
−1
(1 + r1 ) =
α
A2 Y 1
αβ
1+αβ
Y1
α
(1 + αβ)
(144)
which simplifies to:
(1 + r1 ) =
A2
1−α
Y1
(αβ)α (1 + αβ)1−α
β
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Figure 18: General equilibrium effect of a rise in future productivity.
Equation (145) shows that the equilibrium real interest rate will rise in response to a
rise in future productivity A2 , and fall in response to a rise in current productivity A1 (which
exogenously affects current output Y1 ). Put differently, the equilibrium real interest rate
rises with (a rise in) productivity growth.
The general equilibrium effect on investment of a rise in future productivity is illustrated
in Figure 18. A rise in future productivity A2 (holding A1 constant) induces a rise in current
investment (as in Figure 17) for any given real interest rate r1 . In equilibrium, however, the
real interest rate will rise (as per equation (145)) exactly enough to offset the excess demand
for investment, and current investment will remain constant.
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Part 3
The New Keynesian Model
Monopolistic Competition, Sticky Prices and
Monetary Policy
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13
The New Keynesian model
So far our analysis of the economy, and of fiscal policy therein, has been based on a model
where markets are perfectly competitive and prices (of goods, labor and credit) adjust instantaneously to clear markets. These analytical features are sometimes referred to as “neoclassical.” In this section we introduce two main departures from the previous models. First,
we assume that goods markets are imperfectly competitive. Each firm is assumed to hold a
slight degree of monopoly power in the production of a differentiated variety. The presence
of market power is a necessary condition for each firm to have the ability of setting its own
price. Second, nominal goods prices do not fully adjust instantaneously. Both features are
realistic elements of the data.
Systematic and normative analysis of monetary policy. Some of the key concepts we are going to analyze concern the theory and practice of monetary policy. In particular, the NK model will be suitable for the analysis of the systematic component of monetary
policy and of its normative implications. There are two ways (not mutually exclusive) in
which monetary policy can operate. For one, it can be a source of shocks and surprises
for the economy (the unanticipated component). To simplify: in the morning, the governor of the central bank wakes up with a headache and decides, randomly, to increase the
supply of money or reduce the nominal interest rate. The second way, is the systematic
one: the central bank adopts a rule of behavior and (endogenously) responds to the state
of the economy. For instance, if the central bank has the goal of keeping inflation low, it
systematically raises the interest rate when inflation goes beyond a certain threshold, or
when the economy is hit by a rise in the oil price. A natural complement to the systematic
component of monetary policy is the normative component: what should monetary policy
do? How should the central bank respond to a rise in the price of oil, or to a collapse in
asset prices? What criterion should the central bank follow? Our assumption will be that
the goal of the monetary authority will be the one of maximizing households’ welfare. To
address these questions, it is necessary to construct a theoretical apparatus that presents
more explicit foundations in the microeconomic theory of utility and production.
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14
Core with flexible prices
The NK model consists of two nuclei. In the core unit, agents maximize utility and firms
maximize profits in a context in which prices are perfectly flexible. This is our neoclassical
benchmark.27 The combination of this model with the “Keynesian” assumption that goods
prices are substantially rigid constitutes the “synthesis.” The NK model is in fact also referred
to as New Neoclassical Synthesis.28
The economy is populated by three types of agents:
1. A representative consumer who maximizes utility.
2. A large number of firms, each producing a slightly differentiated product and with the
objective of maximizing profits.
3. A central bank.
For simplicity, let us assume that the aggregate quantity of production corresponds to
the total quantity of consumption. This amounts to saying that in this economy there is no
investment, nor consumption by the public sector.
14.1
Utility function with leisure
The representative agent derives utility from the aggregate consumption of all varieties of
goods, C, and disutility from supplying labour. We can write the utility function as:
log C −
1
N 1+φ
1+φ
(146)
where N is the total number of hours worked and the parameter φ, as we will see, is
the inverse of the elasticity of labour supply with respect to the real wage.
The representative household holds an identical equity share in each of the firms. He/she
faces the budget constraint
C = Wr + Γ
27
Better, the benchmark model corresponds to the so-called Real Business Cycle theory. The three basic
ingredients of RBC theory are: 1) agents behave optimally; 2) prices are perfectly flexible; 3) shocks to
productivity are the main driver of the business cycle.
28
For the origins of this definition, see the work of Goodfriend and King (1997)
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where W r is the real wage and Γ are total real profits deriving from the holding the
shares of the monopolistic firms. We abstract here from intertemporal consumption/saving
decisions.
Profits in turn are given by:
Γ = Y − W rN
14.2
Consumption, leisure and labour
A simple principle guides the optimal choice between labour and leisure. We analyze the
marginal behavior of the agent, that is we suppose that the agent is considering whether or
not to work an additional hour. The marginal cost of an additional hour of work is N φ (to
obtain it, it is sufficient to differentiate (146) with respect to N ). On the other hand, by
working an additional hour the agent earns the real wage W r and can consume more. This
guarantees additional utility equal to C1 W r , that is the real wage itself, weighted by the
marginal utility of consumption.
Equalizing marginal cost and marginal utility we get:
Nφ
|{z}
1
Wr
C
| {z }
=
marginal cost of labour
(147)
marginal benefit of labour
Equation (147) defines the labour supply schedule. Indeed, we can rewrite:
N=
Three points are worth noticing:
Wr
C
ϕ1
(148)
• For a given level of consumption, labour supply increases with the real wage.
• For a given level of the real wage, higher consumption corresponds to lower labour
supply. Hence, consumption and leisure are normal goods.
• If consumption and the real wage increase proportionally, labour supply is constant.
14.3
Nominal spending, money and monetary policy
Let us assume that the amount of nominal spending in the economy, P · C, is proportional
to the quantity of money:
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κM = P · C
(149)
where M is the quantity of money, κ is the velocity of money in circulation, and P is the
aggregate level of prices. Equation (149) represents one version of the quantity theory of
money.29 For simplicity, we assume in the equation above that the nominal quantity of money
M already represents the value prevailing at the equilibrium between demand and supply
in the money market. Thus, we assume M to be the instrument of monetary policy. As a
result, any change in M should be considered as a change in money supply. The underlying
hypothesis is that the corresponding money demand adjusts through the variation of the
interest rate to keep the money market in equilibrium at any given period.
Cash in advance. Alternatively, we can think of equation (149) as resulting from
the assumption that the consumer should hold (“demand”) the quantity of money M d in
advance, in order to carry out any purchase:30
M d ≤ κ−1 P C
(150)
Imposing equilibrium in the money market and that constraint (150) is binding yields:
Ms = Md = M
= κ−1 P C
Henceforth we assume that the velocity of money in circulation is constant and equal to 1:
κ = 1.
29
Equation (149) is consistent with additional microfoundations whereby the quantity of money shows
up in the utility function, and the rate of growth of money supply follows an independent and identically
distributed process (see Galı̀ 1999). From the clearing of the goods market Y = C and of the money market
(given the intertemporal choice of consumption, see the next sections) it is possible to derive equation (149).
In such equilibrium, the interest rate is constant. As a result, we can write money demand as:
Md
=
f (i, C)
=
f (i, C)
In the hypothesis of f(·) = 1/k, and ignoring the interest rate because it’s constant, we get
Ms = Md = M =
30
C P
κ
This hypothesis is typically referred to as the “cash-in-advance” constraint.
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14.4
Firms
The production function (identical across all firms) is characterized by constant returns to
scale:
Y =AN
(151)
In particular, A is the level of productivity of labour. Intuitively, if A increases, it is
possible to produce more with the same quantity of labour.
14.4.1
Marginal cost
According to equation (151), 1/A units of labour are required to produce one unit of production (consumption). If the nominal wage is W , the nominal cost to the firm of an additional
unit of labour (i.e. the nominal marginal cost) is:
MC =
W
A
In real terms, we can write:
Wr
W/P
≡
A
A
r
where W is the real wage. Intuitively, the real marginal cost is increasing in the real
M Cr =
wage and decreasing in labour productivity. Moreover, it is worth noticing that, since the
production function exhibits constant returns to scale, the marginal cost coincides with the
unit cost.
14.4.2
Prices and markup
We already mentioned the fact that in this economy there is a large number of firms, each
producing a slightly differentiated product variety. Because of this feature of the market,
each firm has a small monopolistic power on its variety (monopolistic competition). Hence,
firms are not price-takers as they would be in a perfectly competitive market, but they face
a downward-sloping demand function.
In a market characterized by monopolistic competition, the firm sets the price at a
markup m > 1 above its nominal marginal cost:
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P = m MC
W
= m
A
(152)
where W is the nominal wage.
It is important to notice that the optimal markup (that is, the one entailing profit
maximization) is constant:
m = m∗
This follows from the fact that the firm faces a demand curve with constant price elasticity.31
The crucial implication of prices being perfectly flexible is that the firm can set a markup
which is in line with the optimal one, which is constant. This implies that if, for example,
the nominal marginal cost M C in equation (152) increases, the firm also increases its price,
so as to keep m constant and equal to m∗ .
We can summarize in Table 1 below:
Table 1
Markup Pricing Rule
Perfect Competition
m=1
P = MC
m>1
P = m∗ M C
Monopolistic Competition
(flexible prices)
We can link the real marginal cost to firms’ real profits by writing
Γ = Y − W rN
= AN − |A · {z
M C}r N
Wr
= AN (1 − M C r )
1
= AN 1 −
m
Therefore profits are increasing in the markup. In the particular case of perfectly competitive markets (m = 1) profits are zero.
31
Appendix A includes a formal proof.
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14.4.3
Flexible prices and constant marginal cost
An equivalent interpretation of the markup rule under flexible prices is obtained by rewriting
equation (152) as
Wr
A
1
=
m∗
M Cr =
(153)
where M Cr ≡ M C/P and Wr ≡ W/P . We can immediately draw an important conclusion:
in the flexible price scenario, firms are able to keep the real marginal cost in line with its
desired level, and equal to a constant 1/m∗ . This result is central to our analysis, since it
allows us to clearly distinguish a flexible price economy from one (more realistic) with rigid
prices.
There are two possible ways to interpret the profit maximizing condition (153):
1. Labour demand curve.
Recall that, given the production function (151), the
marginal product of labour is given by A. Equation (153) can be written as
A
(154)
m∗
Hence, given the markup m∗ > 1, in a monopolistically competitive market, firms set the
Wr =
real wage below the marginal product of labour. Indeed, equation (154) can be considered
as a labour demand schedule, since it shows to what extent the firms hire labour to align
the real wage to labour productivity. It is instantly clear that the firms are employing a
suboptimal number of workers, since by hiring more labour they could reduce the marginal
product, aligning it with the real wage.
In the particular case in which m∗ = 1, that is under perfect competition, the same
equation implies Wr = A. Hence, in that case, the real wage is exactly in line with marginal
product of labour.
2. Firm marginal cost curve (or supply curve). Substituting Wr from the labour
supply schedule (148) and using the production function (151) we can rewrite the
marginal cost function
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N φC
=
M Cr =
A
1+φ
Y
A
(155)
In general, the marginal cost increases with the level of output (and decreases with the level
of productivity). The reason why the marginal cost increases with the level of output is
simple: in order to increase the quantity produced, marginally, the firm needs to hire more
labour and therefore to increase the real wage it offers.
14.5
Equilibrium employment and output
We can now easily derive analytic conditions that determine the value of employment and
output in equilibrium. In general, we define the natural level of a given variable when
referring to the equilibrium behavior of the same variable under flexible prices.
Natural level of output. Substituting Wr from the labour supply schedule (148)
into (153) we get
1
C Nφ
=
m∗
A
(156)
Using the production function (151)
1
=
m∗
1+φ
Y
A
Solving for Y we obtain a condition for the equilibrium level (or natural level) of output
∗
Y =A
We can notice two things:
1
m∗
1
1+ϕ
(157)
1. For a given constant markup, the natural level of output changes with the level of
productivity.
2. Under monopolistic competition (with flexible prices) the value of the markup is m∗ >
1. This decreases the natural level of output, which would be higher under perfect
competition (when m = 1). Indeed:
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Y perf = A > A
comp.
1
m∗
1
1+ϕ
=Y∗
Monopolistic distortion. The conclusion is that the monopolistic competition introduces a distortion in the equilibrium value of output. This distortion, however, depends
on factors rooted in the goods market. As a result, it cannot be corrected using monetary
policy, but only resorting to long-run policies (fiscal and regulatory) that support higher
efficiency and competition in the goods market (and therefore lower markups).
In order to obtain the natural level of employment, it is sufficient to substitute (157)
into the production function:
N∗ =
1
m∗
1
1+ϕ
(158)
Hence, the natural level of employment is constant and it does not depend on how
productivity varies. This result derives from the fact that the level of productivity affects
the real wage and consumption proportionally. From the labour supply schedule (148), we
see clearly that N is constant.
14.6
The model in compact form
We can now illustrate in a schematic way the two fundamental blocks of our analytic apparatus.
14.6.1
Block 1: Labour market equilibrium
The first block is composed of the determinants of the labour market equilibrium. The latter
depends on the position of two curves in the real wage-employment space (Wr , N ):
• Labour supply schedule Ns , which we can rewrite as:
Wr = N φ C
(159)
The Ns curve defines an increasing relation between the real wage and labour supply
for any given level of consumption.
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• Labour demand schedule Nd given by equation (154) and rewritten as
A
(160)
m∗
Figure 19 illustrates the labour demand schedule (154) and the labour supply schedule
Wr =
(159) in the diagram (Wr , N ). Labour supply Ns is increasing in the real wage for any
given level of consumption. Labour demand is a horizontal line and it corresponds to the
A0
level of productivity m
∗ . The equilibrium is in point E, which corresponds to the natural
level of employment N ∗ represented by the vertical line NN. Recall that the natural level
of employment is constant and independent of the evolution of productivity. Under flexible
prices, the economy is always at a point along the NN line.
Figure 19: Labor market equilibrium under flexible prices
14.6.2
Block 2: Firm supply and aggregate demand
Figure 20 displays the real marginal cost curve MCR of the firm (equation (155)) in the space
(M Cr , Y ). This curve is increasing in the level of output for any given level of productivity.
Notice that in the case of flexible prices, the marginal cost is always equal to the constant
1/m∗ , which is nothing but the inverse of the desired markup.
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Figure 20: Real marginal cost, desired markup and aggregate demand under flexible prices
14.6.3
Flexible prices equilibrium and monetary neutrality
In summary, the following conditions characterize the flexible prices equilibrium:
1
C Nφ
=
m∗
A
(161)
C = AN
(162)
For a given (exogenous) level of A, the system (161)-(162) is a system of two equation in
two unknowns {C, N } (endogenous variables). The production function (151) can then be
used to obtain the level of output Y .
What is then the role of monetary policy (and of the quantity of money in general) in
determining the equilibrium? In particular, what is the role of equation (149), that we write
again below?
M = PC
Such equation is residual to the system, and essentially irrelevant in determining the
equilibrium values of C, N and Y . The key role of equation (149) is to determine the
price level P . In other words, given the equilibrium value of C determined by the system
(161)-(162), the quantity M determines P through equation (149).
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Flexible price equilibrium
given C
| {z }
from the real
block of the model
M
| →
{z P}
:
the nominal quantity M
determines P
This result is known as neutrality of money: M is irrelevant for the determination of
the the real allocations C, N and Y . In other words, the equilibrium of the real side of
the economy is determined independently of the equilibrium of the nominal (monetary) side
of the economy. In other words, under monetary neutrality, there is a complete dichotomy
between the real and the nominal side of the economy.
14.7
Productivity shock: flexible prices
In Figure 21 we analyze the effects of an increase in productivity A in the economy with
flexible prices. The upper panel shows the equilibrium Ns - Nd , while the bottom panel
shows the equilibrium MCR-AD.
The effect of an increase in A is an upward shift of the labour demand curve, from Nd (A0 )
to Nd (A1 ). At the same time, the instantaneous adjustment of prices allows the firms to
keep their markup constant. Hence, the real wage increases to A1 /m∗ and the economy
moves along the labour supply schedule to point B. Point B, however, cannot be sustained
as an equilibrium. Indeed, at the flexible price equilibrium the level of employment should
be constant (equation (158)). This means that the economy should be positioned along the
NN curve. To obtain this, the labour supply schedule has to move. Recall that we draw
the Ns curve in this space for a given level of consumption. As a result, as productivity
increases, consumption will increase proportionally to the real wage, and the Ns curve will
shift up and left. The equilibrium, point E1 on the NN curve, implies a higher level for the
real wage and the level of employment constant and equal to the natural level N ∗ .
A constant markup (under flexible prices) means that also the marginal cost is constant.
Hence, in the bottom panel we represent the effect of the increase in productivity on the
supply curve of the firm. The M CR (A0 ) curve shifts to the right to M CR(A1 ), along
the horizontal line that corresponds to the level of the marginal cost 1/m∗ (recall that the
position of the MCR curve depends on the level of productivity). So the equilibrium moves
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Figure 21: Effects of a rise in productivity under flexible prices.
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from point E0 to point E1 . Analogously, the demand curve shifts rightward. This is possible,
in absence of any response by the monetary policy authority, precisely because prices adjust.
Indeed, for M fixed and equal to M , the decrease in the price level is just enough to allow
consumption, hence income (see equation (214)), to expand. Therefore, the AD0 curve moves
to AD1 . The result is a higher level of output. Hence, a productivity shock determines an
increase in the natural level of output. Since in equilibrium the employment level is constant,
consumption and output increase in a way exactly proportional to productivity.
15
Sticky prices, variable markup and monetary policy
In this section we relax one of the key assumptions that guided the evolution of the economy
up to now, that is the assumption on the evolution of prices. In reality, prices do not
instantaneously adjust in response to the shocks that hit the economy. This happens for a
number of reasons. For instance, because it is costly for firms to collect all the information
necessary to immediately set the optimal price. In practice, firms bear real costs when
adjusting prices (so-called menu costs). Regardless of the causes of price rigidity, we now
proceed to analyze the scenario in which prices are assumed to be temporarily fixed.32
P =P
The crucial consequence of the sticky prices hypothesis is on the cyclical evolution of
the markup. Now, firms are no longer able to instantaneously adjust their price so as to
keep the markup in line with its optimal constant level. We can now write equation (152)
as
W
1
= P
(163)
MC =
m
A
where W
≡ Wr is nothing but the real wage in the case in which prices are rigid (that
P
is, the nominal wage divided by the price level). As a result, the sticky price economy is
characterized by a variable markup m, which fluctuates around the optimal constant value
m∗ .
These fluctuations precisely depend on to what extent prices are sticky. For instance,
we see from equation (163) that the markup increases as productivity increases and that
32
In a more general version of the model, we could assume that prices adjust slowly and gradually. This
would not alter the qualitative conclusions of our analysis.
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it decreases as the real wage increases. The evolution of the marginal cost is exactly the
opposite of the one of the markup.
A new distortion. Notice that the presence of price rigidities introduces a second
distortion (inefficiency) in the economy, that is the deviation of the markup from its optimal
constant value. We can therefore summarize the distortions characterizing the economy as
follows:
1. Distortion linked to the monopolistic competition. The real wage is always below the
level of the marginal product of labour (see equation (154)). The result is that the
employment and output levels are suboptimal. Notice that this inefficiency emerges
both in the perfectly flexible prices scenario and in the sticky prices scenario.
2. Distortion linked to the cyclical evolution of the markup. This distortion is typical
of the sticky prices case. Indeed, firms cannot instantaneously adjust their prices to
stabilize the markup to the constant value that is deemed optimal.33
15.1
Sticky price equilibrium and the role of monetary policy
The hypothesis that prices do not instantaneously adjust radically changes the nature, and
the implications, of the equilibrium. The sticky price equilibrium is defined by the following
system of equations:
A
= C Nφ
m
(164)
C = AN
(165)
M = PC
(166)
For a given level of productivity A, a given level of quantity of money M and price level
P = P , the system (164)-(166) determines the endogenous variables {C, N, m}.
33
In general, the NNS models envisage one additional distortion (that we do not include here), that is the
one directly linked to the costs of inflation. This is the case, for instance, of when we have quadratic costs of
price adjustment or of when the firms can only adjust prices in an asynchronous way. The latter hypothesis
introduces a typical distortion in relative prices.
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The sticky price equilibrium has two main features. First, the markup m is an additional
endogenous variable of the system. Indeed, the presence of sticky prices does not allow the
markup to be always in line with its optimal (and constant) value m∗ . Second, the equation
of the quantity of money is essential in order to determine the equilibrium. In particular,
through equation (166), the quantity of money M determines the level of consumption C.
In other words:
Sticky price equilibrium
given P = P :
{z
}
|
sticky prices
hypothesis
{z C}
|M →
the nominal quantity M
determines C
Combining (164)-(166) we can write:
M φ
A
N
=
m
P φ
M C
=
P A
1+φ φ
1
M
=
A
P
(167)
(168)
(169)
Solving for the markup m:
m=
A
M/P
1+φ
(170)
Hence, in the sticky price equilibrium, the markup responds positively to an increase in
productivity, and negatively to an expansion of the quantity of money. We proceed to
illustrate this result with a graphical analysis.
15.2
Effects of a productivity shock under sticky prices
We can now analyze the effects of a productivity shock under sticky prices. This is represented in Figure 22. The price rigidities alter the evolution of the economy and prevent the
replication of the optimal allocation under flexible prices and constant markup.
We start by analyzing how the labour market operates. In order to understand the
evolution of employment and the real wage we derive logically all the steps.
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Figure 22: Effects of a rise in productivity: sticky prices and no response of monetary policy.
With sticky prices, the markup rises and the marginal cost falls.
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1. While in the flexible price scenario higher productivity requires a fall in prices to
support an increase in demand, in this case the mechanism gets stuck. Prices do not
instantaneously adjust, so consumption does not change. From the production function
equation (151) we obtain:
Y = C = AN
Hence if A increases and C is constant, it must be that N decreases. As a result, the
first point we can establish is that employment falls.
2. Next we derive the effects on the real wage. From the labour supply equation (148)
we can write:
Wr = N φ C
It follows that if C is constant and N decreases (as we established in the previous point),
necessarily the real wage must fall.
Graphically, we represent the fall in employment and the real wage as a shift of the labour
demand schedule Nd , downward and along the Ns curve, which remains fixed (consumption
is fixed given that prices are rigid). The sticky price equilibrium is represented at point E1
and corresponds to a level of employment below potential N1 < N ∗ . Notice that what is key
for the Nd curve to shift downward is that the markup increases more than proportionally to
productivity, so that A/m falls. Therefore, in E1 there is a gap between actual and potential
employment.
The previous analysis helps us understand also the evolution of marginal cost, output
and aggregate demand in the AD-MCR system (see bottom panel of Figure 22).
• The increase in productivity determines a rightward shift of the MCR curve, and an
increase of the natural level of output. The latter level of output would be the one of
point E2 , consistent with the evolution of the economy under flexible prices. However,
in the presence of price rigidities, point E2 cannot be reached. For sure, firms would like
to produce more (increasing their profits) as productivity increased. Yet they cannot
do so because, when prices are sticky, consumption (aggregate demand) is fixed. As a
result, the level of output is constant.
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• At the same time, we know that the real wage falls and the markup increases. Hence,
it must be the case that the real marginal cost (which is the inverse of the markup)
decreases. The sticky price equilibrium is therefore at point S, with a constant output
and lower marginal cost.
Let us define the output gap as the difference between the actual level of output and its
natural level
X ≡Y −Y∗
It is clear that, since at point S output is below its natural level, there is a negative
output gap, and equal to Y0 − Y1∗ , with Y0 = Y1 (since the level of actual output is constant).
In general, it is important to highlight one feature of the sticky price economy. Unlike
in the flexible prices scenario, when prices are rigid the output level is determined by the
demand side, since it is precisely the rigidity of consumption that prevents the expansion of
production that firms would desire.
15.2.1
Optimal response of monetary policy
It is immediately clear that the sticky prices equilibrium of point S is not the optimal one.
In fact, firms would like to expand their production as their productivity is higher (and
their marginal cost is lower) but, due to the rigidity of prices, demand does not increase
accordingly. As a result, price rigidity is a distortion that prevents the economy to enjoy the
full benefits of a productivity shock. In this context, monetary policy can play an active role
to remove such distortion. This is true because aggregate demand depends on the stance
of monetary policy (see equation (214)). With rigid prices, the central bank can expand
the quantity of money available in the economy (and therefore reduce the interest rate) to
support aggregate demand until it becomes high enough to match the level of supply desired
by the firms.
Figure 23 illustrates these dynamics. For simplicity, let’s start from the description of
the equilibrium in the MCR-AD diagram. When prices are sticky, and in the absence of
any monetary policy response, the economy would move from point E0 to point E1 . From
here, if the central bank were to expand money supply, the AD curve would move to the
right, reaching point E2 . At this point, consumption and output are higher and exactly
equal to their natural level. The markup m and the marginal cost stay constant at their
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Figure 23: Effects of a rise in productivity and optimal response of monetary policy.
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optimal level. In other words, despite price rigidity, the response of monetary policy allows
to replicate the equilibrium characterized by flexible prices and constant markup.
It is now easy to have an intuition of what is happening in the labor market. Consider
the Nd − Ns system. If monetary policy allows to replicate a flexible price and constant
markup equilibrium, the real wage has to increase, causing the Nd schedule to move upward.
In turn, the ensuing increase in consumption (recall that consumption increases as a result
of the monetary expansion, and not because of the adjustment of prices) allows the Ns curve
to move, reaching point E2 . Notice that this is in line with what happens in the labor market
in the case of flexible prices, as discussed above when referring to Figure 21.
The scenario we just analyzed frames the role of monetary policy in a completely new
way. In this context, the central bank responds in endogenously to the shock hitting the
economy (in this particular case, a productivity shock), and aims at replicating the same
allocation that would be attained if prices were flexible. In other words, monetary policy
has the objective of compensating for the lack of price flexibility, that is it has the role of
neutralizing one of the two distortions in the economy, i..e, price rigidity. With respect to
the second distortion affecting the economy (i.e. monopolistic competition), monetary policy
is however powerless, since such distortion depends on the structure of the goods market,
which can only be addressed, potentially, by fiscal and/or regulatory policies.
Inflation, output gap and price stability. Notice that, as the economy moves from
point E0 to E2 , the level of output is in line with its natural level. Therefore, a monetary
policy aimed at stabilizing the markup is nothing but a policy of zero output gap. Then,
prices are constant, since even if the firms were able to adjust them downward, they would
choose not to do so. Indeed, following the productivity shock, it is monetary policy that
takes on the task of inducing the necessary expansion of demand. In other words, monetary
policy works in the direction of making the rigidity of prices a non-binding constraint for
firms. It is precisely in this sense that we can conclude that monetary policy removes the
distortion associated to price rigidity.
To better understand this fundamental point, it is useful to draw a comparison with
the previous case, in which monetary policy was passive by assumption. In the previous
case, when the central bank did not expand money supply, the impossibility of adjusting
prices was perceived by the firms as a constraint: the firms were willing to adjust their prices
downward, so as to induce an expansion of consumption which would match the boost in
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production. In this case, however, the monetary authority responds endogenously to the
productivity shock, and keeping prices constant is for the firms an equilibrium behavior : it
is the optimal course of action. This happens because, as the central bank expands money
supply, it creates precisely the expansion of consumption that is necessary to induce the
firms to choose to keep their markup (hence their prices) constant.
The conclusion is that, in the presence of an optimal monetary policy, firms will rationally choose to keep their prices unchanged, and so the optimal inflation rate (i.e. the
optimal rate of change in prices) is necessarily zero. Hence, our theoretical framework provides a rigorous explanation (and one that is based on microfoundations) of why central
banks choose to set price stability (and/or low inflation) as their objective. In the language
of the NK model, this corresponds to a so-called markup smoothing policy, that is a policy
of stabilization of the cyclical fluctuations of the markup.
Pro-cyclical monetary policy. Another result that is worth highlighting is that optimal
monetary policy is pro-cyclical. This is in stark contrast with the traditional view of the
Keynesian stabilization policies (monetary and/or fiscal policies) which typically associates
macroeconomic policies to the concept of counter-cyclical stabilization. In the traditional
view, indeed, the economic cycles are regarded as undesirable. Instead, according to the NNS
perspective, it is desirable that the economy takes full advantage of the productivity shock,
precisely because the agents maximize their respective objective functions (utility function for
the consumer and profit function for the firms). Monetary policy should therefore encourage
this process, by working with the goal of achieving a combination of zero inflation and zero
output gap. In this way, the monetary authority guarantees that the utility of the economic
agents is maximized.
15.3
Effects of a monetary expansion
Let us now analyze the effects of monetary expansion. In this case, it is the central bank that
decides, in an exogenous way, to increase the supply of money. Notice that this experiment
(typically studied in introductory models such as the IS-LM model) describes a role of
monetary policy which is opposite to the one we just discussed. Indeed, the monetary policy
itself is now the source of the exogenous shock, while previously the monetary authority
responded endogenously to changes in productivity.
Consider Figure 24 first, which depicts the flexible price scenario. The quantity of
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Figure 24: Effects of an exogenous increase in the quantity of money: flexible prices.
money increases from M0 to M1 , with M1 > M0 . In the MCR-AD system, if prices did not
adjust, the economy would move to point E1 . However, such point cannot be sustained as
an equilibrium, since under flexible prices the constant markup condition, m = m∗ , should
always be satisfied. Hence, the equilibrium is still in point E0 . It is possible to achieve this
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through an instantaneous adjustment of prices, from an initial level of P0 to the new level
P1 (P1 > P0 ); the price change should be large enough to exactly keep the real quantity
of money, and with this consumption, unchanged. Therefore, under flexible prices, the
monetary expansion is neutral on consumption and production. This can be seen also in
the Ns − Nd system. Since consumption does not change, the Ns curve remains fixed. In
turn, the Nd curve is also fixed, since (for an unchanged level of productivity) the markup is
constant, and at its desired value m∗ . The equilibrium is therefore unchanged at point E0 .
Figure 25 illustrates the effects of a monetary expansion in the case of sticky prices. In
the MCR-AD diagram, the equilibrium moves to point E1 . This is possible because, with
sticky prices, the markup falls below its optimal constant level m1 < m∗ . The AD curve
shifts rightward. With unchanged prices, P0 = P , the real quantity of money increases:
(M1 /P0 ) > (M0 /P0 ). At the new equilibrium in point E1 , the level of output Y increases
from Y0 = Y ∗ to Y1 > Y0 . Since the natural level of output Y ∗ is unchanged, the monetary
expansion causes the output gap Y1 −Y ∗ to increase. The MCR curve is in the same position,
because the level of technology is unchanged by assumption.
Let us now consider the Ns -Nd diagram. The expansion of consumption produces a
shift to the left of the Ns curve (for the same given level of the real wage). At the same
time, the fall of the markup shifts the Nd curve upward. We can therefore notice that, under
sticky prices, the labor demand curve moves also in response to variations in the markup,
even when the level of productivity stays unchanged: this brings about an increase in the
real wage. If the prices are permanently fixed at the level P = P0 , the new equilibrium will
be in point E1 , with an employment level of N1 > N ∗ , which is above the natural level of
employment.
15.4
The Phillips curve
We have seen so far that, in the case of rigid prices, the behavior of the markup (or of the
real marginal cost) varies radically depending on whether or not monetary policy responds
optimally. We can distinguish two cases:
1. Optimal monetary policy. When monetary policy allows to replicate the behavior of
the economy under flexible prices, the markup is always constant and firms do not
have any incentive to change their prices. At the same time, the output gap is always
zero. This equilibrium implies zero inflation and output gap.
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Figure 25: Effects of an exogenous increase in the quantity of money: rigid prices.
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2. Suboptimal monetary policy. When monetary policy does not behave according to the
optimal prescriptions (i.e., it does not react in order to keep the markup constant), we
know that a positive productivity shock generates a negative output gap and causes
the markup to increase. In this case, firms would have an incentive to change their
prices (in particular, they would like to lower them). Suppose that prices were only
partially rigid. If this were the case, we would observe a (maybe partial) fall of prices
and a negative output gap.
From the latter discussion we can deduce that the model embeds a positive relationship
between the output gap and inflation: the larger the deviation of output from its natural
level, the larger inflation. As we know, this relationship is typically called the Phillips
Curve.34
We will now formally derive the Phillips curve. Recall that under sticky prices (and
with a suboptimal monetary policy) the real marginal varies according to:
1+φ
Y
M Cr =
A
It is convenient (also for future reference) to rewrite the relationship above in logarithmic
form:
mc = (1 + φ)(y − a)
(171)
where mc ≡ log(M Cr ) , y ≡ log(Y ) and a ≡ log(A). In particular, the marginal cost
fluctuates around its desired (or natural) level which is given by the constant 1/m∗ . Hence,
we can define the deviation of the marginal cost from its natural level (in logs)
mc
f ≡ mc − (−µ∗ )
where µ∗ ≡ log(m∗ ). Considering equation (171) we have
34
Notice that positive inflation can always be interpreted as positive deviation from zero, which is nothing
but the natural level of inflation.
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mc
f = (1 + φ)(y − a) + µ∗
µ∗
= (1 + φ) (y − a) +
(1 + φ)
µ∗
= (1 + φ) y − a −
(1 + φ)
∗
= (1 + φ) [y − y ]
(172)
where y ∗ ≡ log (Y ∗ ) and Y ∗ is the natural level of output given by equation (157). As a
result, equation (172) states that the deviations of the real marginal cost from its natural
level are proportional to the output gap (that is to the deviation of output from its natural
level).
15.4.1
A more general theory of price setting
For the sake of simplicity, we worked so far under the extreme assumption that prices were
rigidly fixed. To derive the final formulation of the Phillips curve, however, we will now refer
to a more general theory of how firms set their prices. According to such a theory, inflation
has two main drivers. The first has to do with the role of the marginal cost and therefore of
the output gap, as we saw above. The second justifies a role for inflation expectations.
Role of the marginal cost and of the output gap. When the markup and the
marginal cost are not constant (hence when the output gap is not zero), firms would like
to adjust their prices in proportion to the deviation of their marginal cost from its desired
level. Hence, there exists a positive relationship between inflation and real marginal cost.
Equation (172) suggests that this, in turn, can be written as a relationship between inflation
and the output gap.
Price lottery and expectations. Let us now assume that prices are not rigidly fixed,
but that they can be adjusted only occasionally. In particular, the ability of adjusting prices
is determined by the outcome of a lottery. When productivity increases (or decreases), only
the firms that draw the lucky ticket can adjust their price, while the others are forced to
stick to the price previously chosen. What is the optimal course of action for a firm that
draws today a lucky ticket? It is reasonable to look at the (deviation of the) marginal cost
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from its natural level today. But there’s more to it. The firm that today is lucky, does not
know whether it will have the same lucky strike also in the future, hence it does not want to
set a price today that is too distant from a general average of prices. As a result, if the firm
expects inflation to be high in the future, it will tend to set a relatively higher price today,
exploiting the lucky outcome of the lottery.35
The two arguments above suggest that the determinants of current inflation are two: (i)
the marginal cost gap; and (ii) expected future inflation. Formally:
π = π e + γ mc
f
(173)
π = πe + λ x
(174)
where γ is a parameter measuring the elasticity of inflation with respect to the marginal cost
gap. Using equation (172) and defining the output gap as x ≡ y − y ∗ we can write the final
formulation:
where λ ≡ γ(1 + φ) is the slope of the Phillips curve.
16
Interest rate, expectations and new AD curve
So far we have hardly discussed the role of the interest rate in the economy. As a matter of
fact, we assumed for simplicity that money supply was the instrument of monetary policy.
In reality, the central banks implement and announce their monetary policy actions in terms
of changes in the nominal interest rate (or short term interest rate).
To understand the role played by the (real) interest rate in the determination of the
optimal expenditure choice of the consumer, imagine that the economy (as in our previous
intertemporal model) only exists for two periods. For a given level of the real interest
rate, and for a given time schedule of income, the consumer will chose the optimal path of
consumption across the two periods, taking into account the following budget constraint:
35
The example of the lottery is certainly ad hoc, but it clearly illustrates why a firm would want to take
into account future conditions when setting its price today. We would get a similar behavior assuming that
there exists quadratic (convex) costs of price adjustment. Since in that case the cost of adjusting prices
is larger the higher the rate of price change, the firm will take into account future conditions in costs and
demand to avoid having in the future a price which is too far away from its desired level, therefore entailing
a very high cost of adjustment.
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1
1
C 2 = Y1 +
Y2
(175a)
1+r
1+r
In equation (175a), 1 + r is the real interest rate. Hence, the left-hand side represents the
C1 +
present discounted value of consumption, while the right-hand side is the present discounted
value of income.
Let us write the intertemporal utility of consumption as
log C1 + β log C2
where β < 1 is the intertemporal discount factor. The optimal profile of consumption is
summarized by the following efficiency condition:
C2
= β(1 + r)
(176)
C1
Intuitively, the intertemporal profile of consumption depends both on the intertemporal
rate of substitution and on the real interest rate. In the particular case in which β(1+r) = 1,
consumption is constant over time. The intuition of equation (176) is simple, and it reflects
the optimal allocation of consumption/savings across the two periods. Suppose that the
agent has to choose whether to consume an additional unit of the good today or tomorrow.
The alternative to consuming today is that of postponing consumption until tomorrow, hence
saving and lending the additional unit at an interest rate of 1 + r, which means being able to
consume 1 + r more units of consumption tomorrow. In utility terms, the marginal cost of
foregoing an additional unit of consumption today 1/C1 should be set equal to the marginal
benefit of consuming 1 + r more tomorrow, weighted by the future marginal utility 1/C2 and
discounted using the factor β. From this point of view, the interest rate represents the price
of future consumption relative to current consumption.36
36
Mathematically, equation (176) can be obtained solving the budget constraint for C2 :
C2 = −C1 (1 + r) + d
where d ≡ (1 + r)Y1 + Y2 is the present value of income. Plugging C2 into the utility function:
log C1 + β log(−C1 (1 + r) + d)
The first order condition with respect to C1 implies:
1
1
− β (1 + r) = 0
C1
C2
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16.1
Natural rate of interest
According to our NK framework, we have to distinguish the behavior of the interest rate
in the case of flexible price (the natural level) from what prevails under price rigidity. In
particular, the natural interest rate is to be interpreted as the level of interest rate which
evolves independently of monetary policy.
From equation (157), recall that we can write the natural level of consumption as
∗
∗
C =Y =A
1
m∗
1
1+ϕ
(177)
In equation (177), the only element varying over time is productivity A. Hence plugging
(177) into equation (176) we obtain
∗
(1 + r ) = β
−1 A2
1
(m∗ )− 1+ϕ
1
A1 (m∗ )− 1+ϕ
A2
= β −1
A1
(178)
As a result, we see that the interest rate depends on two factors: (i) the discount factor
β; and (ii) the rate of growth of productivity A2 /A1 . Let us consider two cases:
• When productivity is constant over time (A1 = A2 ), the interest rate is also equal to
a constant
(1 + r∗ ) = β −1
Substituting into equation (176) we obtain
C2 = C1
In other words, if productivity is constant, the representative agent wishes to keep the
profile of consumption constant. In this event, indeed, the benefit of borrowing (that is, the
real interest rate) is exactly equal to the discount factor 1/β, which represents the rate of
impatience of the consumer. Hence, the return that the agent yields from borrowing (and
substituting consumption intertemporally) is exactly compensated by the impatience which
favors consuming today.
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• Increasing productivity: A2 > A1 . In this case, the interest rate is given by β −1 (A2 /A1 ).
Substituting into (176) yields
C2
A2
=
(179)
C1
A1
Therefore the rate of growth of consumption coincides with the rate of growth of productivity. Intuitively, if the agent expects productivity to be higher in the future, he/she
desires to borrow today in order to consume more tomorrow. This drives the real interest
rate up until the agent is satisfied with a profile of consumption that exactly matches the
profile of growth of productivity. According to this interpretation, the natural interest rate
moves to restore the equilibrium in the credit market.
16.2
Interest rate, monetary policy and aggregate demand
Intuitively the real interest rate deviates from its natural level when prices are sticky. Unlike
our earlier analysis, when monetary policy acted directly by varying the money supply, we
can re-interpret monetary policy actions in terms of variations of the interest rate, to then
analyze the effects on consumption expenditure.
Let us stick to the two-period intertemporal dimension. Suppose, for simplicity, that
the agents expects consumption, output and the markup to be at their natural level in the
second period. In other words, assume that in the short run (the current period) prices are
temporarily rigid, while they go back to full flexibility in the second period (or long run).
So the level of consumption in the second period is given by the natural level
1
1 1+ϕ
m∗
Recall that the natural level of consumption cannot be influenced by monetary policy. Since
C2∗ = A2
consumption at time 2 is exogenous from the point of view of the central bank (because
it depends on the desired level of the markup m∗ and on productivity at time 2), we can
rewrite equation (176):
β −1 ∗
C
1 + r1 2
" 1 #
β −1
1 1+ϕ
=
A2
1 + r1
m∗
C1 =
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Hence, in the short run (which coincides with the period in which prices are rigid)
monetary policy can have an impact on aggregate demand. In particular, it can cause the
real interest rate to increase, determining a fall in consumption. Intuitively, a higher level
of the interest rate corresponds to an increase opportunity cost of consuming today (instead
of tomorrow), that is it increases the price of current with respect to future consumption,
which makes current consumption go down.
Notice that the central bank does not have direct control of the real interest rate, but
only of the nominal one. In this particular case, this distinction disappears since the real
interest rate in period 1, 1 + r1 , is given by the nominal interest rate adjusted for expected
inflation in period 2
1 + r1 =
1 + i1
1 + π2e
However, the expectations are for the economy to evolve according the flexible price equilibrium in period 2, hence π2e = 0, implying that 1 + r1 = 1 + i1 .
The above result establishes an important principle. When monetary policy displays a
level of credibility sufficient to anchor inflation expectations at zero, it maximizes its ability
to have an impact on the real interest rate by changing the nominal interest rate. Also notice
that the key condition allowing a relationship between consumption and the interest rate to
emerge is precisely that prices are sticky in the first period. If this was not the case, that is
if prices were perfectly flexible in every period, consumption would match its natural level
both in period 1 and in period 2, evolving independently of monetary policy.
16.3
Interest rate gap and new IS curve
The previous discussion is useful two introduce two new elements. The first one is an
aggregate demand curve, which is written as a relationship between the output gap and the
real rate; this corresponds to a microfounded version of the traditional IS curve. The second
element is the definition of the interest rate gap, meant as the deviation of the real rate from
its natural level, which we described above.
Consider a logarithmic version of equation (176) for any given pair of periods:
c = ce+1 − (r − ρ)
(181)
where ce+1 is expected consumption in the following period, ρ ≡ − log β and r ≃ log(1 + r).
Moreover, recall that because of the goods market equilibrium y = c we can write
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y = x + y∗ = c
where x = y − y ∗ is the output gap. Plugging c into equation (181) we obtain
∗
x = xe − (r − ρ) + ∆y+1
(182)
∗
∗
where ∆y+1
≡ y+1
− y ∗ . In particular, from (157), we can write
∗
∆y+1
= ∆a+1
In turn, from (178) we can write the natural interest rate in logs as
r∗ = ρ + ∆a+1
∗
∗
from which it results that ∆y+1
= r∗ − ρ. Plugging ∆y+1
into (182) we get
x = xe − (r − r∗ )
(183)
= xe − re
where re ≡ r−r∗ represents the deviation of the real interest rate from its natural level. Hence
the curve (183) directly relates the current output gap to its future expected value, and it
inversely relates it to the real interest rate gap. There are three fundamental differences with
respect to the traditional IS curve.
• Higher output in the future means higher output today. We can explain this by making
reference to the concept of consumption-smoothing which is embedded in equation
(176). When the consumer expects income to be higher in the future, he prefers to
maintain its path of consumption virtually unchanged, therefore he starts to consume
more even today. This expectational effect increases the level of current demand and
of current output.
• The negative impact of the real interest rate on the current level of output, instead,
reflects the principle of intertemporal substitution of consumption that we described
above. When the interest rate is higher, the consumer prefers to save and to invest the
proceeds in (real or financial) investment activities that today seem relatively more
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attractive. As a result, the consumer prefers to postpone consumption until the next
period, when the return on investment will grant higher income. As a matter of fact,
when the interest rate increases, the consumer substitutes consumption today with
consumption tomorrow. Hence, the interest rate is the relative price of current with
respect to future consumption.
• Notice, in particular, that the reference variable is the interest rate gap. When the real
interest rate moves in line with its natural level (that is re = 0) we have that the path
of output (and therefore of consumption) is in line with the natural path, exactly as
equation (179) portrays.
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Appendix A
Profit Maximization under Flexible Prices
Suppose to analyze the profit maximization problem of a generic firm of the ones producing in the market. Denote this representative firm with the index j. The firm faces a
downward-sloped demand curve for its product, with a price elasticity of ϑ :
Y (j) =
with ϑ > 1.
P (j)
P
−ϑ
Y
(184)
In particular, Y (j) is the demand for the variety produced by the firm j, Y is the
aggregate demand in the economy, P (j) is the price of the variety produced by firm j and P
is the general price index. Intuitively, the demand curve (184) is characterized by both an
income and a substitution effect. Indeed, the demand Y (j) for the good j increases when the
general demand for all the goods in the economy Y increases (income effect). Moreover, the
demand Y (j) is inversely related to the relative price, that is the price of the given variety
j with respect to the average price in the economy. The relevance of this substitution effect
is measured by the price elasticity ϑ.
The problem of the firm is that of choosing its price P (j) and labor demand N (j) in
order to maximize its profits:
P (j)
W
Y (j) − N (j)
(185)
P
P
subject to the constraints given by the demand function (184) and by the production function
Y (j) = AN (j)
(186)
Substituting N (j) from (186) and Y (j) from (184), the problem of the firm becomes that of
choosing only the price P (j) to maximize
P (j)
P
1−ϑ
W
Y −
P
Y
A
−ϑ
W
Y
+ϑ
P
P
P (j)
P
P (j)
P
−ϑ !
The first order condition is
(1 − ϑ)
P (j)
P
−ϑ−1
Y
=0
AP
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This simplifies into
P (j) =
1
1 − ϑ1
W
= m MC
A
(188)
where
ϑ
≡ m∗
ϑ−1
is the constant value of the markup. This expression coincides with equation (152) in the
m≡
ϑ
precisely because the
text. Hence we see that the markup coincides with the constant ϑ−1
price elasticity ϑ is constant.
We can rewrite the markup expression as:
m∗ =
1
1 − ϑ1
We can then notice that the markup decreases as the price elasticity of demand increases:
↑ϑ
↓ m∗
In particular, when the elasticity ϑ tends to infinity, the markup tends to its limit value of
1.
[ϑ → ∞] → [m∗ −→ 1]
In other words when the demand curve is infinitely elastic (that is, horizontal), the
markup is equal to 1 and the pricing condition (188) becomes identical to the one of the
perfectly competitive market:
P (j) = M C
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Bibliography
Goodfriend, M. (2004) “Monetary Policy in the New Neoclassical Synthesis: A Primer,”
Federal Reserve Bank of Richmond Economic Quarterly, Vol. 90/3.
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Part 4
Rules and Discretion in a Model of Inflation
Targeting
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17
Rules and discretion in policy-making
A central issue in the theory of monetary policy is the one related to the desirability of
conducting policy according to given rules. For instance, a policy which adopts an inflation
target, or a money supply growth rate target on an interest rate target. In this section we
will try to understand what are the incentives for the monetary policy authority to comply
with such rules, once they are in place. It will be clear that a key concept in the interaction
of the central bank with the economic agents is the of credibility.
Modern macroeconomics points out that the behavior of some key variable such as
inflation, output and the interest rates essentially depends on the expectations that economic
agents hold regarding the future stance of policy (monetary or fiscal). If, for instance,
monetary policy follows a systematic rule (that is, a rule which is stable over time and that
can be perfectly anticipated by the agents), the macroeconomic equilibrium can be correctly
deduced from the expectations of how the monetary authority will react in a number of
possible future circumstances (e.g., in the event of an oil shock). But the key issue is the
following: what guarantees that, after agents have formed their expectations, the central
bank will act consistently with what they expect? If the central bank has some room
for discretion after expectations have been formed, what ensures that it will confirm those
expectations? And therefore, anticipating this, will the rational agents take into account such
incentives of the central bank? What are the consequences on the welfare of the economic
agents of a policy authority which decides under discretion, i.e., an authority that is allowed
to reconsider its promises once made? What are the possible remedies?
18
A model for the analysis of monetary policy under
inflation targeting
In this section we present a simple model to analyze the inflation targeting policy of the
central bank. The choice of focusing on inflation targeting is motivated by two main reasons:
(1) it is a paradigmatic example of monetary policy conducted on the basis of a specific rule;
(2) it is a monetary policy regime which is by now common in many countries, both in
advanced and in developing economies.
The model has two components:
• A Phillips Curve (or aggregate supply curve) which relates the current inflation rate
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to the future inflation rate and the output gap;
• A monetary policy rule, derived from the maximization of an explicit objective function
of the central bank.
18.1
The Phillips curve
The first component of the model is a Phillips curve which is augmented by expectations.
Define π as the inflation rate and x as the output gap, that is the deviation of current
output from its potential level. The potential (or natural) level of output is defined as the
level of output prevailing under flexible prices. As a result, the necessary condition for the
output gap to be different from zero is that the economy is characterized by some kind of
rigidity which prevents the instantaneous adjustment of prices (for example, menu costs).
The drivers of the cyclical dynamics of the natural level of output are, by definition, outside
the control of monetary policy. For instance, possible drivers are technological advances or
fiscal policy.
We can write the aggregate supply curve as:
π = π e + λx + u
(190)
In equation (190), π e stands for the expectation regarding the level of inflation in the next
period, which is formulated using information available today; u is an inflationary (or costpush) shock and λ > 0 is a parameter measuring the slope of the curve in the (π, x) space for
a given level of the expectations. In general, we can think that the slope λ is inversely related
to the intensity of the price rigidity in the economy: the more intense the price rigidity, the
lower the inflation, and therefore the lower the effects on inflation of the variation of the
output gap. We define (190) as the PC curve.
As illustrated in our previous sections, the presence of inflation expectations in equation
(190) is reasonable in an economic environment in which prices are not flexible, and are
adjusted based on the expectations on the future value of costs and demand (a forwardlooking behavior). Firms adjust their prices only occasionally. When they do get the chance
to adjust the price, they anticipate the fact that they will probably have to stick to the newly
set price for a number of periods, and therefore they take into account today the expected
future rate of inflation. This is done to avoid the erosion of their relative price (that is the
price of their product with respect to the average level of prices in the whole economy). As
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a result, in equation (190), for any given level of the output gap, expected future inflation
determines a higher level of current inflation.
The fundamental implication of including inflation expectations in the PC curve is that
they make the inflation rate today depend on the expectations of how monetary policy will
be conducted in the future. As we will see, the ability of the central bank to affect such
inflation expectations is a key element for the evaluation of alternative monetary policies.
18.2
Monetary policy rule
The second component of the model is the monetary policy rule. We assume that the central
bank, in every period, has the objective of maximizing the following welfare function:
−
1
(π − π T )2 + ω (x − x∗ )2
2
(191)
where π T ≥ 0 is the inflation target, x∗ ≥ 0 is the optimal (desired) level of the output gap,
and ω > 0 is a parameter describing the weight of the output gap relative to inflation in the
objective function of the central bank.37 As a result, the central bank wishes to minimize the
deviations of inflation from its target, but also the deviations of the output gap from a given
level which is considered to be optimal.38 Notice that in this formulation the loss function
is quadratic, implying that positive and negative deviations of the output gap from their
respective targets are both penalized in a symmetric way. In other words, an inflation rate
which is 2 percent above the target is penalized exactly as an inflation rate which is 2 percent
below the target (and this works analogously for the output gap). From the maximization
of the objective function, we can derive several alternative monetary policy rules, which we
will discuss in detail in the next sections.
19
Discretionary policy and inflation bias
Starting from the foundational work of Kydland and Prescott (1977), much of the modern
theory of monetary policy has focused on the incentives faced by the policy authorities
h
i
2
Notice that the expression 12 (π − π T )2 + ω (x − x∗ ) is a loss function, since it penalizes deviation of
inflation and the output gap from their desired values. With a minus sign in front, this expression becomes
a welfare function.
38
In the language of the most recent literature of monetary policy, which can define this as a policy of
“flexible inflation targeting.”
37
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when setting their instruments. Suppose that the economic agents (firms and/or workers)
specify today the contents of their wage contracts (or product prices). An essential element
to consider while making their decision is the expectation of the future level of inflation.
Indeed, when the agents set the level of the nominal wage for the next few years, they
run the risk that, in the future, inflation will erode the purchasing power of their wage.
Suppose that the central bank commits today to keep inflation equal to zero. Of course, this
announcement will have an impact on the wage salaries that are negotiated today.39 At this
point, the sequential timing of the decisions is key. In fact, once the agents have formed their
expectations (and fixed the level of the wage), what is the incentive of the central bank to
make good on its promise and act to keep inflation at zero? And furthermore: if the agents
anticipate already today that the central bank could have some incentive to deviate from
its announcement of zero inflation, will they incorporate this expectation in the prices and
wages that they set?
In the present section, we analyse the interaction between the agents and the central
bank under the assumption that the central bank acts in a regime of discretion. In other
words, the central bank, in principle, is free to set its monetary policy stance once the
agents’ expectations have already been formed. We will show that, in this context, a wellknown inflation bias problem will appear: the equilibrium level of inflation is higher than
the one that would prevail had the central bank been forced to act in line with its original
announcement. This situation testifies of how a monetary policy authority (just like any
other economic policy authority) with the most virtuous intentions (i.e. maximizing the
welfare of the society), but suffering from a credibility deficit, can lead the economy to an
“undesirable” equilibrium.
Assumptions. Our model is based on three key assumptions.
1. Sequence of events. The monetary policy decision is made after agents have formed
inflation expectations.
2. Discretion. The central bank acts under discretion. This means that there is no
mechanism which makes the policy announcement issued at the time in which agents
39
Although the example of the wage contract is likely to be the most relevant, we can extend the same
reasoning to price setting. The expectations of future inflation (and therefore of the future stance of monetary
policy) are important, since high future inflation would erode the relative position of the price that the firm
sets today with respect to the future general average of prices.
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formed expectations strictly binding.
3. Output distortion. This assumption constitutes a necessary condition for the validity
of our analysis, since it determines the ex-post incentive of the central bank to deviate
from the announced course of action. We assume that desired level of the output gap
is different from zero and equal to a positive value:
x∗ > 0
This assumption has the implication that the central bank does not aim at exactly
replicating the level of potential output (i.e, x = 0), which is the level that would prevail
under flexible prices. A possible rationale for this assumption is the presence of distortions
in the economy, which render the natural level of output lower than its optimal level. Such
distortions may stem from the presence of a tax on labor or a market structure with monopolistic competition (which prevents the replication of the production and employment level
under perfect competition). However, there could also be non-economic reasons inducing the
central bank to prefer a level of output above the natural one. For instance, there could be
political pressures affecting the behavior of the central bank and limiting its independence.
In this case, the political establishment could be prone to supporting economic expansions
in order to maximize its probability of being re-elected.
For simplicity (and without loss of generality) we will assume henceforth that the inflation target is zero (π T = 0) and we will consider a completely deterministic environment (in
which shocks are absent):
πT = 0
u = 0
19.1
The problem of the central bank
The sequence of events is as follows:
1. The central bank announces a given inflation target π T , which we assume equal to zero
for simplicity.
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2. Agents form their expectations regarding inflation (and, potentially, they set wages
and prices on the basis of those expectations).
3. For a given level of expected inflation, the central bank sets the level of the output gap
and, through the Phillips curve (190), the inflation rate.
We can therefore write the problem of the central bank as follows:
1
∗ 2
2
max − ω (x − x ) + π
2
(192)
subject to the constraint
π = π e + λx
Substituting x from equation (190) we can write the objective function as:
"
2 #
π−f
1 2
π +ω
− x∗
max −
2
λ
(193)
where we defined
f ≡ πe
The term f can be regarded as exogenous to the maximization problem. We can do this
precisely because we assumed that in every period the central bank can act with discretion
and re-optimize. Indeed, the central bank has no ex-ante constraint (commitment) on how
it should design future monetary policy. This is equivalent to assuming that inflation expectations are predetermined with respect to the decisions of the central bank, consistent with
our timing assumptions. The policy authority, therefore, can choose the level of the output
gap (and hence of inflation) taking expectations as given (in this way, the central bank can
potentially choose a level of inflation different from what agents conjectured).40
The first order condition of problem (193) implies:
π+ω
π−f
− x∗
λ
1
=0
λ
40
(194)
This solution method corresponds to the so-called logic of backward-induction. This is to say that the
game between the agents and the central bank is solved starting from the decision of the central bank (step
2) which considers expectations as already determined, and then work our way backward.
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Alternatively:
ω
(x − x∗ ) = 0
(195)
λ
The intuition for condition (195) is simple: it states that at the optimum the marginal
π+
cost of inflation and its marginal benefit must be equalized. To understand the marginal
cost and benefit of inflation, suppose inflation were to increase by one unit. This has a direct
marginal cost given by M Cπ ≡ π, that is the derivative of the loss function with respect to
π. But higher inflation is also a cost because it increases the inflation gap. From the Phillips
curve (190) we deduce that an additional unit of inflation generates (1/λ) more units of
output gap (for a given level of inflation expectations). After weighting this for its impact
on utility, which is given by ω, we obtain ωλ (x − x∗ ), representing the cost of higher inflation
expressed in terms of a larger output gap. Preceded by a minus sign, −(ω/λ) (x − x∗ )
indicates the marginal benefit of higher inflation. Equalizing marginal benefit to marginal
cost we obtain equation (195).
OP curve. Plugging (195) into the PC curve (190) we obtain (always assuming that
u = 0):
π=
ω
ω + λ2
[λx∗ + π e ]
(196)
Equation (196) describes the inflation rate optimally set by the central bank for any given
desired level of the output gap x∗ and for any given level of inflation expectations. Formally,
we can define this as the reaction function of the central bank. The latter observes inflation
expectations held by the agents, and sets the optimal current inflation rate given that value
of expectations. Two observations are worth noting regarding the optimal inflation rate:
(i) it is increasing in the desired level of the output gap; (ii) it increases with inflation
expectations.
We label equation (196) as OP curve (from optimal policy). This curve determines the
relationship between current and expected inflation in the (π, π e ) space. At each point on
the OP curve, marginal cost and benefit of inflation are equalized - that is, equation (195)
is satisfied). A key result is already clear. Suppose that inflation expectations were equal to
zero, that is π e = 0. Equation (196) immediately reveals that actual inflation (which results
from the interactive game between the agents and the central bank) is not zero, rather it is
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Figure 26: Curve OP: equilibrium inflation chosen by the central bank for any given value
of inflation expectations.
ωλ
∗
positive and equal to π = ω+λ
2 x > 0. Why is that the case? Can this be an equilibrium of
the game, consistent with the incentives of both the agents and the central bank?
19.1.1
Graphical analysis
Figure 26 depicts the curve OP, with π e on the x axis and π on the y axis. We also chart
the 450 line, the locus of the point in which actual and expected inflation coincide. Notice
that the OP curve is upward-sloped, although its slope is less than 1:
ω
<1
ω + λ2
Hence, every increase in expected inflation results in a less than proportional increase
0<
in actual inflation. Why? On the one hand, if expected inflation increases, the central bank
should increase actual inflation one-to-one if it wants to keep the level of the output gap
constant (see the Phillips curve). However, there is a cost associated to the rise of actual
inflation, hence for the central bank it is optimal to increase inflation less than proportionally.
In other words, as we have seen already, the central bank should consider the marginal
benefits and marginal costs of higher inflation.
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Equilibrium. In Figure 26, the point where the OP curve crosses the 450 line corresponds
to the equilibrium inflation rate π
e. This is the level of inflation such that actual and expected
inflation coincide. Why should actual and expected inflation coincide in equilibrium? This
amounts to assuming that agents have rational expectations. In short: actual and expected
inflation cannot diverge for too long. Indeed, if agents observe that actual and expected
inflation diverge systematically, they will revise their expectations in line with this. As a
result, in the long run, it must be that:
π = πe
(197)
In summary, the equilibrium point should be where two conditions are satisfied: (i) marginal
benefits and marginal cost are equalized (hence the point must lie on the OP curve); (ii)
condition (197) holds. Plugging (197) into equation (196) we obtain an analytical expression
for the equilibrium level of inflation:
ω ∗
x >0
(198)
λ
The crossing of the OP curve and the 450 line is not just informative of the fact that, in
π
e=
equilibrium, expected and actual inflation should coincide. Indeed, it also indicates that
this happens for a positive level of inflation, that is π = π e = π
e > 0. Why π = π e = 0
cannot be an equilibrium, even if actual and expected inflation coincide? Suppose that the
central bank was to announce zero inflation, so that π e = 0. In reality, this announcement
cannot be credible. In fact, at zero inflation, the marginal benefit of inflation exceeds its
marginal cost. To understand this, notice that the marginal benefit of inflation is given by:
e
∗
M Bπ = − ωλ (x − x∗ ) = − ωλ π−π
−
x
. At zero inflation we have that:
λ
ωx∗
>0
(199)
λ
On the other hand, the marginal cost of higher inflation is given by M Cπ=0 = π = 0. Hence
M Bπ=0 =
it holds:
M Bπ=0 > M Cπ=0
(200)
Given this, the agents will not believe the zero inflation announcement. Actually, anticipating that, at the margin, the central bank will have an incentive to generate inflation
after expectations are formed, they will revise up their expectations, until the central bank’s
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marginal cost and marginal benefit of higher inflation are aligned. In turn, this implies setting higher prices (or wages) today. The result of this will be an increase in the equilibrium
level of inflation equal to π
e > 0 in equation (198).
Notice that the equilibrium level of inflation π
e depends on three factors:
1. It increases with the desired level of the output gap: ↑ x∗ ↑ π
e. In Figure 26 this
corresponds to a vertical movement of the OP curve, since the rise of x∗ makes the
y intercept of the curve increase. The intuition is simple. The higher x∗ , the higher
the incentive of the central bank to generate inflation, since the marginal benefit of
inflation, M Bπ (see (199)), is higher.
2. It is increasing in the weight assigned to the output gap in the objective function: ↑ ω
↑π
e. In Figure 26 this corresponds to both an upward shift of the OP curve, and to an
increase in its slope. Analogously to what just described above, the rise of ω causes
the marginal benefit of inflation to increase, since the central bank’s objective function
assigns relatively more weight to the output gap.
3. It is decreasing in the slope λ of the Phillips curve: ↑ λ ↓ π
e. In Figure 26, a rise of λ
has an impact both on the slope and on the intercept of the OP curve. The effect on
the slope is negative, while the effect on the intercept is ambiguous.41 The net result
is that the equilibrium level of inflation falls. Intuitively, λ measures the trade-off
between inflation and the output gap as expressed by the Phillips curve (190). That
is, how much more inflation it is necessary to bear in order to increase the level of
the output gap (for a given level of expectations). The higher is λ, the lower is the
marginal benefit of inflation (see (199)).
19.2
Inflation bias and credibility
The fact that the equilibrium level of the inflation rate is positive when x∗ > 0 is a key result
in monetary policy theory. This suggests an explanation of why, in advanced economies, we
41
∗
ωλx
To understand this point, it is enough to obtain the derivative of the intercept ω+λ
2 with respect to λ:
ωx∗ (ω + λ2 ) − 2λ2 ωx∗
(ω + λ2 )
2
=
ωx∗ (ω − λ2 )
(ω + λ2 )
2
Hence, as λ increases, the intercept increases, stays constant or decreases depending on whether ω > =
< λ2 .
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observed high inflation rates for extended periods of time, in particular during the 1970s and
1980s. At the heart of this result lies the problem of credibility of the central bank.
To understand this point, consider again the interaction “game” between the central
bank and the economic agents. At the start, agents form their inflation expectations and set
prices (and wage contracts) accordingly. Suppose the agents expect zero inflation (π e = 0),
maybe because the central bank released some announcement which hints at the fact that
in the future it is willing to fight inflation vigorously. The key issue here, as we already
mentioned above, is the following: is the announcement of the central bank credible? In
other words: is it optimal for the central bank, ex post, to ratify the expectations of zero
inflation? We already provided a negative answer to this question, arguing that in this case
the marginal benefit of increasing inflation exceeds the marginal cost.
A different way to illustrate the same point is the following. From equation (190) we see
that, given expected inflation π e = 0, zero actual inflation (π = 0) can only be achieved if
the output gap is equal to zero (x = 0). However, from the objective function (192), clearly
this is not optimal for the central bank, which can do better by setting a positive level
for the output gap, x = x∗ > 0. As a result, for a given level of inflation expectations, the
central bank has an incentive to set an output level which is higher than the natural (thereby
generating inflation). In turn, since the agents have rational expectations, they recognize exante (that is, when setting prices and/or wages) the ex-post temptation of the central bank.
As a result, they instantaneously incorporate the expectation of higher inflation, since they
know that the central bank will have the incentive to deviate ex-post from its announcement
of zero inflation.42 In equilibrium, the level of inflation that the public expects (thus also the
actual level of inflation for any given level of the output gap in equation (190)) will increase
until it equalizes the central bank’s marginal cost and benefit of increasing output above its
natural level. The outcome of this strategic interaction is a positive inflation rate level π
e>0
in equation (198).
Plugging (198) into equation (195), we obtain the equilibrium solution for the output
gap:
x
e = x∗ −
λω ∗
x =0
ωλ
(201)
42
At this point, it is clear why we solved for the equilibrium level of inflation positing π = π e . In fact, it is
not possible for the central bank to systematically fool the agents, who rationally incorporate the expectation
on the future behavior of the central bank when today they set prices.
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Hence we see that the inflation rate π
e > 0 is the one necessary to make it optimal for the
central bank to choose an equilibrium level for the output gap which is equal to zero. In
other words, π
e > 0 is precisely the inflation rate which equalizes marginal cost and marginal
benefit of higher inflation, hence it eliminates the incentive of the central bank to push
output above its natural level.
One aspect is immediately clear. Even if this result is an equilibrium of the game played
by the agents and the central bank, it does not correspond to the optimal solution, which
would instead be the one of point (e
π = 0, x
e = x∗ ). In such a case, in fact, the loss would
amount to L = 0. Instead, in the case of discretion, plugging (201) and (198) into the loss
function (191) we obtain
ω 2
L =
+ ω (x∗ )2 > 0
λ
d
Notice the paradox. Even if the central bank has the objective to maximize the welfare
of society (as described by the objective function (191)), the economy only ends up with a
cost in terms of positive inflation and no benefit in terms of output gap.
19.2.1
Commitment: solution to the credibility problem
What are the possible solutions to the inflation bias problem? The analysis above clearly
implies that the key word here is credibility. Namely, the central bank should make sure that
agents believe its zero inflation announcement. The obvious way to achieve that goal is to not
allow the central bank to choose every period its favorite combination of inflation and output
gap after the agents’ expectations are formed. In other words, the central bank should be
able to commit to keep inflation equal to zero in a binding way. If agents anticipate that the
central bank will not be allowed to re-formulate its decisions ex-post, they will immediately
align their expectation to the target announced by the monetary authority.
In practice, there are various ways to make commitment operational. For instance, by
electing a conservative central banker, with a strong anti-inflationary reputation (Rogoff
1985). Alternatively, the economy could decide to enter a fixed exchange rate system (just
like Italy in the EMS ), therefore relinquishing the temptation of systematic devaluations
of its currency (Giavazzi and Pagano, 1991). Alternatively, it would be possible to define a
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transparent and credible regime of inflation targeting. In the next sections, we will analyze
more in detail the differences between commitment and discretion.
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20
Stabilization bias and response to shocks
Recent research on the credibility issue of monetary policy provided evidence that this problem is even more pervasive than what was believed until a few years ago. Indeed, the
analyses have shown how discretion can lead to an inferior outcome even when we relax our
third assumption, namely that the desired level of the output gap is positive (x∗ > 0). In
a stochastic context (i.e., when the economy is hit by shocks and the central bank should
play a role in stabilization), even in the absence of any inflation bias, the behavior of the
central bank under discretion is suboptimal, since it involves a suboptimal response to the
same shocks (stabilization bias).
We introduce two variations with respect to our previous analysis.
1. We assume that it is optimal for the central bank to replicate the level of output that
would prevail under flexible prices. This implies doing away with the standard source
of inflation bias (i.e. the temptation of the central bank to push output above its
natural level, that is the one consistent with flexible prices), which we identified in the
analysis above.
Formally, we assume:
x∗ = 0
(202)
2. We introduce the exogenous shocks u. Therefore, we remove the assumption that u = 0
in every period.
20.1
Problem of the central bank in the presence of inflationary
shocks
Under assumption (202), we can write the problem of the central bank as:
1
(π − π T )2 + ωx2
2
subject to constraint (190). Plugging x from equation (190), the objective function becomes:
"
2 #
1
π
−
f
u
max −
(π − π T )2 + ω
2
λ
max −
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where in this case the exogenous term from the point of view of the central bank also includes
the exogenous shock u:
fu ≡ π e + u
The first order condition of the problem above implies that, in every period, the marginal
benefit of varying the output gap should be equal to the marginal cost of letting the inflation
rate deviate from its target:
ω
(203)
π − πT + x = 0
λ
Equation (203) determines the optimal monetary policy rule. We label the above relationship between inflation and the output gap equation MPR schedule (monetary policy rule).
Equivalently, we can write:
π = π T − αx
(204)
where
ω
λ
is the slope of the MPR schedule. Notice that this slope is affected by two factors:
α≡
• the relative weight of output in the loss function, ω.
• the slope of the Phillips curve, λ, which measures the inflation cost which should be
born in order to increase the output gap by a given percentage.
Equation (203) describes the combination of inflation and the output gap which, for a
given target π T , minimize the loss function (191). The rule has an intuitive prescription:
when inflation is above its target (π > π T ) the central bank should intervene so as to generate
a negative output gap, namely x < 0. In other words, if inflation exceeds its target, it is
necessary to make output decrease below its potential level. The contraction of the output
gap, through the supply curve (190), works against the rise of inflation. In brief, we can
name this policy as leaning against the wind.
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Figure 27: The system PC-MPR: Phillips curve and monetary policy rule.
20.2
The PC-MPR system
Equations (190) and (204) define a system of two equations in two unknowns, (x, π). The
intercept of the PC curve is given by the term π e , while the slope is given by the value of
the parameter λ (we assume that the value of the shock is initially u = 0). The same figure
represents the (204) relationship as negatively-sloped schedule, with intercept π T .
The system is illustrated graphically in Figure 27:
• PC:
π = πe + λ x + u
• MPR:
π = π T − αx
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20.3
Equilibrium and convergence to the target
Figure 27 represents the relationship (190) in the (x, π) plane, for a given level of inflation
expectations. Suppose that initially the system is in point E0 . Notice that here the PC and
MPR curves cross, hence (x0 , π0 ) is an equilibrium, and both the Phillips curve PC and the
monetary policy rule MPR are satisfied.
20.3.1
Short-term equilibrium
This initial point corresponds to a negative output gap x0 < 0. The reason why the central
bank is willing to accept a negative output gap is very simple. As you can see, when the
system is in point E0 , it must necessarily be that π e > π T . Hence, inflation expectations
exceed the inflation target of the central bank. Formally, we can solve for x0 by imposing
P C = M P R:
πT − πe
x0 =
α+λ
Therefore, π e > π T implies x0 < 0. Notice that at E0 the inflation level π0 is somewhere
between the target and the expectations:
π T < π0 < π e
The fact that π0 < π e is consistent with x0 < 0. Indeed, from equation PC, we can derive
π0 − π e = λx0 < 0
20.3.2
Long-term equilibrium
Point E0 in Figure 27 cannot be a long run equilibrium though. In fact, overtime agents will
persistently observe that π < π e , and they will re-formulate their expectations. Namely, they
will gradually revise downward their inflation expectations. This will trigger a fall of π e and
therefore a gradual shift of the PC curve (from PC0 to PC1 ), since its intercept depends on
π e . The revision of expectations will lead to a gradual rise of the output gap. Convergence
will stop in point E1 , where it holds:
π1 = π T = π e
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At the same time, in point E1 , we will have that x1 = 0. Moreover, notice that, in the
particular case in which the inflation target is zero, the long term equilibrium E1 corresponds
to π1 = π T = 0.
20.4
A reduction of the inflation target
We illustrate in Figure 28 the effects of a reduction of the inflation target. Suppose that
the economy is initially located at the long run equilibrium E0 , where the inflation target
is π T > 0. Here, π0 = π T = π e . Suppose now that a new governor of the central bank is
appointed, who decides to revise downward the inflation target to the level π T = 0. At the
beginning, the effect will be a downward shift of the MPR curve. Indeed, a reduction of
π T corresponds to a decrease in the intercept of the MPR schedule (that is, a change of the
central bank’s preference). Point E1 however is a short term equilibrium. Here, the economy
is in a recession: x1 < 0. Notice that the economy moved along the PC curve. Therefore,
point E1 represents a situation in which inflation expectations have not changed yet.
As we already mentioned, point E1 is only a short term equilibrium, where it holds:
π T < π1 < π e
As a result, inflation expectations will gradually adjust downward, causing the PC curve to
move from its initial position P C0 to its long run position P C1 . The long term adjustment
involves a gradual rise of the output gap. In the final equilibrium point E2 , expectations are
in line with the new target and the output gap is again zero:
π T = π2 = π e
x2 = 0
20.5
A change of inflation expectations
Figure 29 depicts the effects of a rise of inflation expectations.
Suppose that, even if the inflation target of the central bank has not changed, agents
suddenly expect that inflation will rise in the future. For instance, because of a sequence of
wage hikes, or as a result of the devaluation of the country’s exchange rate. The economy is
initially in point E0 , with π T = π0 = π e = 0. An increase in inflation expectations determines
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Figure 28: A reduction in the inflation target
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Figure 29: A rise of inflation expectations
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a shift to the left of the P C curve. The short term equilibrium is in point E1 , with π1 > π T ,
and with x1 < 0. Notice that in the short term a rise of inflation expectations, within
an inflation targeting regime, has a contractionary effect on the output gap. The latter
is crucially determined by the fact that the PC moves along the MPR curve. Namely, the
negative output gap is the optimal response of monetary policy to the sudden rise in inflation
expectations.
The long run dynamics are once again the result of the interaction between expectations
and the inflation target. At point E1 we have:
π T < π1 < π e
Inflation expectations cannot diverge from the inflation target for too long. Therefore,
in the long run, expectations will move downward, gradually bringing back the PC curve
to its initial position. In brief: a sudden increase in inflation expectations, under inflation
targeting, determines a temporary period of negative output gap. Notice that the extent to
which the central bank is credible can have an impact on the process of convergence. Indeed,
we can suppose that the process of convergence will be faster the more credible the inflation
target π T = 0 .
20.6
Supply shock and preferences of the central bank
Recall that the slope α of the MPR curve is positively related to the relative weight of
the output gap in the loss function and negatively related to the slope of the PC curve.
We depict in Figure 30 two different schedules, M P Rα1 and M P Rα2 , with respective slope
α1 > α2 . Hence, the steeper curve, MPRα1 , corresponds to the case in which the volatility
of the output gap is assigned a relatively higher weight than in MPRα2 .
Consider now the case of a positive supply shock: u > 0. Suppose that the shock only
hits in the current period, and reverts back to zero right after. This shock shifts the PC
curve upward, from P C0 to P C1 : to any given level of the output gap is now associated to
a higher value of inflation. In other words, inflation rises for reasons that are unrelated to
the value of the output gap. In the short run, the central bank will react to the shock. We
study the reaction of the central bank precisely by looking the intersection of the PC and
MPR schedules. The central bank will respond to minimize the rise of inflation by reducing
the output gap. However, by doing so, the central bank cannot avoid reducing the output
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Figure 30: Supply (cost-push) shock and central bank’s preferences.
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gap to a value below zero.
It is important to notice that it is unfeasible for the central bank to instantaneously
replicate the (optimal) combination of zero inflation and zero output gap. If inflation rises
for reasons which are exogenous to the output gap, every attempt at bringing inflation
back to zero will necessarily lead to pushing the output gap below zero. In other words,
in response to a supply shock, the central bank faces the dilemma (trade-off ) between how
much inflation and how much recession (negative output gap) it should bear. The crossing
point of the PC and MPR curves is the optimal solution to this dilemma (a more detailed
analysis will follow).
Hence. the result of the response of monetary policy is a combination of higher inflation
and a negative output gap. It is evident that the short run equilibrium inflation rate depends
on the slope of the MPR curve. In the case of the M P Rα1 curve, the short run equilibrium
is point E1 , while in the case of the M P Rα2 curve the short run equilibrium is E2 . In
particular, inflation will rise more when the MPR curve is steeper (MPRα1 curve, with slope
α1 ). In other words, a supply shock causes higher inflation when the central bank has a
relatively stronger preference for stabilizing the output gap. Viceversa, the fall of the output
gap is less sizeable in the case of the MPRα1 curve.
Notice that both the P C0 and the P C1 curve are drawn for the same, and given, level
of inflation expectations π e = 0. This depends on the temporary nature of the shock. Since
the shock hits the economy only in the current period, the agents know that it will revert to
zero in the future, and therefore do not modify their expectations.43
20.7
Monetary policy trade-off and algebraic solution
The previous example of a supply shock clearly illustrates the trade-off faced by the monetary
authority. To better understand this issue, let’s derive the algebraic solution of the model.
From the MPR curve, we can write:
x=−
Plugging into the PC equation:
1
π − πT
α
43
Later on, we will see that monetary policy can play a role here. Indeed, by recognizing that it can have
an impact on expectations, the central bank will be able to minimize the costs associated to the supply
shock.
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π = πe −
Rearranging:
λ
π 1+
α
λ
π − πT + u
α
= πe +
λ T
π +u
α
(206)
At this point it is convenient to adopt a slightly different notation to denote expectations.
It is useful to distinguish the behavior of each variable in each time period. For a generic
period t = 0, 1, 2, ..., we define Et πt+1 as the expectation on the inflation rate in period t + 1,
formulated on the basis of information available in the current period t. Suppose we are in
period t = 0. At time “zero,” we write equation (206) as:
π0 =
αλ
α
α
E 0 π1 +
πT +
u0
α+λ
α (α + λ)
(α + λ)
(207)
Notice that π T is a constant, therefore it does not have any time index. Our goal is
that of rewriting the equation above eliminating the term which incorporates expectations
E0 {π1 }. At time t = 1 we can write the same equation as:
π1 =
αλ
α
α
E1 {π2 } +
πT +
u1
α+λ
α (α + λ)
(α + λ)
(208)
Taking expectations with respect to time t = 0:
α
αλ
α
E0 {E1 π2 } +
πT +
E 0 u1
α+λ
α (α + λ)
(α + λ)
αλ
α
E0 {E1 π2 } +
πT
=
α+λ
α (α + λ)
α
αλ
=
E 0 π2 +
πT
α+λ
α (α + λ)
E0 {π1 } =
(209)
(210)
In the equation above we proceeded in a few steps:
• The second equality is justified by the fact that, since the shock u is only temporary,
agents expect that the shock will revert back to zero tomorrow: E0 u1 = 0.
• By taking expectations of a constant we obtain the constant itself, meaning E0 π T = π T .
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• The third equality holds because of the law of iterated expectations, stating that
E0 {E1 π2 } = E0 π2 . In other words, what an agent expects today (that is, at t = 0)
to expect tomorrow (t = 1) regarding what will happen two periods ahead (t = 2), is
exactly equal to the expectation today of what will happen in two periods.
At this point equation (209) gives us an expression for E0 π1 which we can plug into
equation (207). This yields:
π0
2 !
α
α
α
=
πT +
+
u0
α+λ
α+λ
(α + λ)
2
α
α
λ α
α
1+
πT +
u0
E 0 π2 +
=
α+λ
αα+λ
α+λ
(α + λ)
α
α+λ
2
λ
E 0 π2 +
α
(211)
Notice that up to now we only iterated for two periods, t = 0, 1, 2. This explains the exponent
“2” in the first two terms of equation (211). If we were to iterate further, for instance up to
time 4, we would obtain the following:
π0 =
α
α+λ
4
E 0 π4 +
and so forth. Notice:
λ
α+λ
1+
|
α
α+λ
+
α
α+λ
{z
2
+
α
α+λ
3 !
}
πT +
α
u0
(α + λ)
4
5
α
α
• The further we look into the future, the smaller the number α+λ
, α+λ
, etc.., since
α
< 1. Hence we can assume that after a sufficient number of periods, let’s say j
α+λ
j
α
E0 πj = 0.
periods, we have α+λ
!
2 3
α
α
α
• Moreover, the expression 1 +
+
+
+ .. is a geometα+λ
α+λ
α+λ
{z }
|
1
α+λ 44
ric series, converging to the number 1− α = λ .
( α+λ )
We can now simplify and rewrite (211) solving for the inflation rate:
44
In general, recall that if |a| < 1 it holds:
∞
X
j=0
aj =
1
1−a
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α
u
α+λ
Using the equation of the MPR, we then obtain the output gap solution:
π = πT +
1
u
α+λ
From (212) and (213) we can make a few observations:
x=−
(212)
(213)
1. As we discussed graphically with the aid of Figure 30, a positive supply shock increases
inflation, while reducing the output gap. Equivalently, in response to a cost-push shock
u, inflation and the output gap move in opposite directions. This behavior of inflation
and the output gap is directly linked to the trade-off faced by the central bank. Indeed,
for any given level of output gap, a shock to u increases inflation. Vice versa, for any
given level of inflation, a shock to u determines a fall of the output gap. As a result,
if the central bank aims at neutralizing higher inflation, it should reduce the output
gap. But this would only worsen the initial fall of the output level.
2. Equations (212) and (213) describe the optimal reaction of the central bank when the
objective is that of minimizing the loss function (191). In particular, equations (212)
and (213) suggest that, in response to a cost-push shock, in general, it is not possible
to instantaneously reach the point where (π = π T , x = 0). In fact, this allocation can
be achieved only in two circumstances:
• u = 0 =⇒ In absence of cost-push shocks there is no trade-off for the central bank
• ω = 0 =⇒ α = 0. If the weight of the output gap in the loss function is zero, the
central bank only aims at reaching a point where π = π T . This can be achieved
without cost, whatever the value of the output gap.
3. The previous result bears an important implication for monetary policy. In the absence
of cost-push shocks, if the central bank keeps output in line with its natural level (that
is, x = 0 in each instant), it is feasible for the central bank to keep the inflation rate
at zero.
4. Since the presence of cost-push shocks makes it unfeasible for the central bank to
implement the optimal allocation (π T , 0), inflation will deviate from its target for
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a period of time which is proportional to the duration of the shock. In this case,
the inflation targeting regime can be regarded as a monetary regime in which the
convergence to the target can only happen gradually.
20.8
Expectations and optimal response to the shock
Although the case in which the central bank acts under discretion is probably the most
realistic one, we show in this section that the response to inflationary shocks derived from
equations (212) and (213) is not the optimal one. The key point here is the ability of
the central bank to affect expectations. Figure 31 plots the PC curve together with the
indifference (iso-loss) curves of the central bank. Each of these curve depicts combinations
of inflation and the output gap which correspond to the same level of loss, according to
equation (191). The further away from the origin, the higher the loss. Under the assumption
that the inflation target is zero π T = 0 , the point (0, 0) is the optimum.
In the absence of shocks, the economy is initially located at the long run equilibrium
(0, 0). Under the assumption of target zero, in that point we have that π e = 0. The point
corresponds to the Phillips curve P C0 . Suppose now that a temporary inflationary (costpush) shock hits the economy. Hence, the shock will disappear in the following period. Let
us analyze the response of the central bank in two cases:
1. Discretion. In this case the PC curve shifts upward to P Cdis . The short run equilibrium is at point DIS, with a combination of inflation and output gap given by (πdis ,
xdis ). It is important to highlight that the curve PC dis corresponds to an unchanged
level of inflation expectations, π e = 0.
2. Commitment. Can the central bank improve on point DIS? The only way to do this,
in response to the shock, is to manage to shift the PC curve downward, towards point
COMM. What determines the position of the PC curve? Of course the shock but,
crucially, also inflation expectations. For a given value of the supply shock, the only
way to reach point COMM is to somehow generate expectations of deflation, meaning
π e < 0. Hence, the PC curve corresponding to a value of the shock u = 1 and which
contemporaneously incorporates expectations of deflation is curve PC comm. Clearly
this curve is tangent to the indifference curves at a point closer to the origin, namely
in correspondence of a lower inflation level.
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Figure 31: Optimal response to a cost-push shock: commitment vs discretion.
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In response to a cost-push shock, how can in practice the central bank generate expectations of deflation? The only way to achieve that is by manipulating the response of the
output gap. Figure 32 plot the evolution of inflation and output gap under discretion (solid
line) and under commitment (dashed line), in response to a temporary cost-push shock.
Under discretion, inflation increases and the output gap falls. Since inflation expectations are unchanged at π e = 0 (by definition, expectations cannot be affected in a regime
of discretion), inflation increases in the first period to then revert to zero when the shock is
exhausted. Under commitment, the path of the two variables is different. In the first period,
the output gap falls. However, suppose now that the central bank follows a different strategy: it announces today that, notwithstanding the shock reverting back to zero tomorrow,
it will keep the output gap tomorrow at a negative level. Suppose also that, by definition of
commitment, agents deem this strategy credible. As a result, inflation expectations adjust
accordingly, becoming negative. To put it differently: after observing the cost-push shock,
it is optimal for the central bank to announce today a path for the output gap which is more
contractionary than it would be under discretion. This strategy allows the central bank to
manage expectations in an efficient way. Precisely taking advantage of the fact that, according to the PC curve, current inflation depends on the expectations of future inflation, the
response of inflation today is way less sizable under commitment than it is under discretion.
Notice that the impact response of inflation under commitment is lower than in the case
of discretion:
π0comm < π0dis
This is precisely because of the effect on expectations. Under commitment the PC curve
will incorporate expectations of negative inflation, which are immediately mirrored by the
current behavior of inflation. Thus, while under discretion inflation reverts back to zero in
the period right after the shock, in the case of commitment inflation remains at a negative
level for a while. The dynamics of the output gap are consistent with those of inflation.
Notice that in the case of commitment the initial fall of the output gap is smaller. This
happens because, under commitment, managing expectations in an optimal way leads to
facing a less strict trade-off. Under commitment, from the PC curve, it is necessary to
reduce output less to generate any given reduction of inflation.
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Figure 32: Dynamic path of output gap and inflation in response to a cost-push shock:
commitment vs. discretion.
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20.8.1
Optimal policy and time inconsistency
Figure 32 shows an additional remarkable aspect of monetary policy under commitment.
Consider the situation of the economy in period 1. Under commitment we have:
π1comm < 0
xcomm
< 0
1
We can now focus on the utility function of the central bank and ask: evaluating utility
from the point of view of time 1, is the central bank acting in the optimal way? What should
the central bank choose in order to maximize welfare in period 1? We can write welfare in
period 1 under commitment as follows:
f comm = − 1 (π comm )2 + ω (xcomm )2 < 0
W
1
1
1
2
It is clear that by choosing π1 = x1 = 0 the central bank could increase welfare at time
1. The choice π1 = x1 = 0 is precisely the one characterizing the regime of discretion.
However, didn’t we establish that under commitment monetary policy could achieve higher
welfare? The key point to understand is that commitment corresponds to the inter-temporal
maximization of welfare, which depends not only on utility at time 0 (that is, when the
shock hits the economy), but also at time 1 and onwards.
Limiting our horizon to the perspective of time zero is therefore myopic. However, doing
so is tempting for the central bank that chooses the regime of commitment. The latter,
indeed, constrained itself at time 0 by announcing that at time 1 it will not immediately
push inflation and the output gap to zero. When period 1 arrives, the central bank could
be tempted to renege on its promises. In fact, the choice (π1comm < 0, xcomm
< 0) to which
1
it committed at time zero does not look as attractive through the lens of time 1. From the
viewpoint of time 1, it would seem optimal to choose (π1 = 0, x1 = 0) in order to minimize
the intra-temporal loss function. We formally refer to this temptation as time inconsistency.
What is optimal for the central bank to choose at t = 1 (on the basis of a plan initially laid
out at t = 0) does not look like optimal anymore if we take the perspective of t = 1, rather
than of t = 0.
Hence, we can clearly see the crucial importance for the monetary authority of adopting
a credible form of commitment at time t = 0. In our example the commitment constraining
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the authority should be sufficiently credible to induce agents to believe that, when time 1
arrives, the central bank will not fall into the temptation of deviating from what it announced
at time zero.
21
Aggregate demand and interest rate rule
So far, we have been describing monetary policy only in terms of the allocation of inflation
and of the output gap. In fact, we could have interpreted inflation as the objective of
monetary policy and the output gap as its instrument. In reality, the central bank uses the
short term nominal interest rate as its own instrument, and takes advantage of the aggregate
demand transmission channel to influence the output gap and inflation.
21.1
New AD curve
In the model described so far, the PC and MPR curves represent, respectively, the aggregate
supply curve and the monetary policy rule. In order to complete the description of the
economy, we should formalize the aggregate demand side. Using the derivation of the new
IS curve, the formulation we now adopt is a generalization of the IS curve (which inversely
links the output level to the interest rate) along two directions: (i) to take into account the
role of expectations, (ii) to distinguish between real and nominal interest rate:
x = xe − φ [(i − π e ) − r∗ ] + g
(214)
In equation (214), i is the nominal interest rate, xe is the expected output gap, r∗ is the
natural level of the interest rate and g is a demand shock (for example a shock to government
spending). Hence equation (214) relates the current output gap positively to the future
expected output gap, and negatively to the real interest rate.45
21.2
Optimal interest rate rule
Notice that the relevant interest rate for the consumption decision is the real rate, r ≡ i−π e .
The central bank, instead, controls the nominal interest rate i, which is the true instrument
of monetary policy. The solutions for inflation and the output gap as a function of the cost45
In the previous section, implicitly, ϕ = 1, as a result of the logarithmic preferences. Moreover, g = 0.
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push shock are available from equations (212) and (213). In the simple case of temporary
and i.i.d shock, implying ue = 0, substituting (213) into equation (214) we can write:
−
1
u = −φ i + φπ e + φr∗ + g
α+λ
recalling that
xe = −
(215)
1
ue = 0
α+λ
α
ue = π T
α+λ
Hence from (215) we can write a rule on the current interest rate, with the equilibrium in
πe = πT +
(212) and (213):
i = π T + r∗ +
1
1
u+ g
φ (α + λ)
φ
(216)
Equation (216) illustrates an important principle: in absence of (demand and/or supply
u = g = 0) shocks, for the central bank it is optimal to set the nominal rate instrument in line
with: (i) the real natural rate, that is the level of the real interest rate which is compatible
with flexible prices (and therefore with zero inflation expectations); (ii) the inflation target.
21.2.1
Optimal rule vs. Taylor Rule
It is interesting to ask what is the relationship between the optimal interest rate rule (even
if derived under the hypothesis of discretion) and a simple Taylor rule. This will help us
understand how the latter rule, which many regard as a possible simple paradigm of monetary
policy, is indeed close to an optimal rule.
Going back to the solution for the inflation rate (212) and substituting u we can rewrite
(216) as:
i = π T + r∗ +
1
1
(π − π T ) + g
αφ
φ
(217)
Equation (215) therefore includes all the elements of a simple Taylor rule, where the
nominal interest rate (in deviation from the natural real rate) is set in response to deviations
of the inflation rate from its target. As a result, the Taylor rule seems in line with the
solution of the optimal monetary policy problem under discretion that we derived above.
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A typical feature of the Taylor rule is the size of the coefficient of response to inflation.
The so-called Taylor principle, indeed, requires that this coefficient is larger than 1. The
intuition is simple. Every increase in the inflation rate (above the target) should be accompanied by a more than proportional increase in the nominal interest rate, so as to induce
the corresponding increase in the real interest rate. As a matter of fact, it is the latter
that influences the consumption and investment decisions. Suppose that inflation increases
above its target. To manage to bring it back in line with the target itself, the central bank
should succeed in influencing the demand for consumption. This calls for an increase in the
real rate. If this was not the case, that is if the increase in the nominal rate was less than
proportional to the increase in inflation, we would have a decrease in the real rate. But this
would support demand for consumption, with the end of reinforcing the initial inflationary
drive, therefore compromising the goal of stabilizing inflation to its target value.
Hence, the Taylor principle requires that the following condition is satisfied in equation
(217):
ω
φ<1
(218)
λ
Condition (218) involves three parameters: ω (the relative weight of the output gap in
αφ =
the loss function of the central bank), λ (the slope of the Phillips curve) and φ (the elasticity
of the output gap with respect to variations of the real interest rate in the new IS curve).
According to empirical estimates, it is plausible that ωλ < 1, and that φ ∼ 1. Hence it seems
reasonable that the optimal course of action of the central bank under discretion satisfies
the Taylor principle.
There is, however, an alternative way to guarantee that the rule (216) (which, recall, is
the solution for the interest rate in presence of the optimal monetary policy) is consistent
with the Taylor principle. Suppose that we add and subtract the term ϕπ (π −π T ) to equation
(217), where ϕπ strictly satisfies:
ϕπ > 1
α
u, from (217) we get:
Recalling that (π − π T ) = α+λ
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1
1
α
u + g − ϕπ
u
φ (α + λ)
φ
α+λ
1
= r∗ + π T + ϕπ (π − π T ) + Φ u + g
φ
i = r∗ + π T + ϕπ (π − π T ) +
(219)
where the coefficient Φ corresponds to
1
Φ≡
(α + λ)
1
− ϕπ α
φ
Rule (219) is in principle equivalent to rule (216). However, it satisfies two principles:
• Since ϕπ > 1 by assumption, rule (219) satisfies the Taylor principle;
• Unlike rule (216), which desribes a relationship between the monetary policy instrument and a hardly measurable variable such as the natural interest rate r∗ , rule (219)
is operational. This means that it indicates how the nominal interest rate should be
set in relation to an observable and measurable variable, i.e., the inflation rate.
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Part 5
Liquidity Traps
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22
The return of liquidity traps
In normal times the central bank has control over the stance of monetary policy: by controlling the nominal interest rate, and to the extent that prices are relatively sticky, it is
also able to control the real interest rate, and therefore the level of aggregate demand in
the economy. An effective monetary control may however not be the prevailing state of the
world at all times. Since the mid 1990s Japan has been locked in a liquidity trap, i.e., a
situation in which extremely low (or zero) nominal interest rates are no longer sufficient to
stimulate aggregate demand. The same situation has prevailed, since 2008, also in the US
and in other industrialized countries.
Keynes (1936) gave the following definition of a liquidity trap: “There is the possibility...that, after the rate of interest has fallen to a certain level, liquidity-preference may
become virtually absolute in the sense that almost everyone prefers cash to holding a debt
which yields so low a rate of interest. In this event the monetary authority would have lost effective control over the rate of interest. But whilst this limiting case might become practically
important in future, I know of no example of it hitherto.”
More recently Paul Krugman, studying the onset of a liquidity trap in Japan in the
early 1990s: “A liquidity trap may be defined as a situation in which conventional monetary
policies have become impotent, because nominal interest rates are at or near zero: injecting
monetary base into the economy has no effect, because [monetary] base and bonds are viewed
by the private sector as perfect substitutes.”
In a liquidity trap, people are indifferent between bonds and cash because the rates of
interest both financial instruments provide to their holder is practically equal. The interest
on cash is zero and the interest on bonds is near-zero. Hence, the central bank cannot affect
the interest rate any more (through augmenting the monetary base) and has lost control over
it. Figure 33 shows that central banks around the world pushed monetary policy short-term
interest rates towards zero with the onset of the Great Recession in 2008. Such a situation
was however prevalent in Japan since the late 2000s
In this section we study the conditions that determine the onset of a liquidity trap.
We show that a liquidity trap is the outcome of a sufficiently large contraction in aggregate
demand. During the Great Recession of 2008-11 such demand contraction came from a
financial market shock and a collapse in house prices that hit the balance sheet of households
and induced them to save more, i.e., to a process of deleveraging.
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Figure 33: Nominal policy rates in different industrialized countries
23
A model of the zero lower bound
So far we have ignored the possibility that the nominal interest rate it might reach the zero
lower bound. We now assume:
it =
it +ζ
|{z}
average
≥ ζ
≥ 0
where ζ ≥ 0 is a term that can be interpreted as the (maturity) term premium. The real
interest rate therefore is also subject to the constraint:
e
rt = it − πt+1
(220)
e
≥ ζ − πt+1
≡ rtmin
The above constraint has important implications on the aggregate demand curve. Let’s
first of all assume (without loss of generality) that there are no productivity shocks, so that
the natural real interest rate reads
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rt∗ = ρ
We will henceforth assume that the term ρ is exogenous and time varying:
ρ = ρt
The demand curve can be written:
e
yt = yt+1
− (rt − ρt )
(221)
The lower bounds constraint (220) on the real interest rate implies also a constraint on
aggregate demand:
e
yt = yt+1
− (rt − ρt )
e
≤ yt+1
− (rtmin − ρt )
e
e
= yt+1
− ζ + πt+1
+ ρt
≡ ytmax
For the sake of simplicity we will ignore the role of future income in the determination of
e
aggregate demand, and assume yt+1
= 0.
Policy rule. Monetary policy obeys the following Taylor-type rule:
e
it = rt∗ + ϕπ (πt − π) + πt+1
(222)
with ϕπ > 1,and where π is the inflation target. The above rule states that if inflation is at
e
the target (πt = π) then the real interest rate it − πt+1
follows the natural real rate rt∗ .
We can rewrite the monetary policy rule:
rt = rt∗ + ϕπ πt
|{z}
e
it −πt+1
where we have assumed for simplicity that the inflation target π = 0.
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(223)
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Discount factor shock. Starting from a zero inflation equilibrium (i.e., πT −1 = 0),
we assume that the shock hitting the economy at time t = T is a shock to the discount
factor ρt (i.e., a shock to the natural real interest rate). Notice that this shock is akin to
an aggregate demand shock. In fact, from the demand equation (221), a fall in ρt lowers
current output for any given value of the real interest rate.
We also assume that the shock is purely temporary and hits the economy only at time
T. Hence:
ρT −1 = 0
ρT = ρbT < 0
ρt = 0 for all t > T
Solving backward. We are going to solve for the equilibrium backward. In other
words by first computing the equilibrium at time t+1 to then infer the equilibrium at time
t. Since the economy is no more in a liquidity trap in T+1 the behavior of monetary policy
can be described as:
rT +1 = rT∗ +1 +ϕπ πT +1
|{z}
(224)
=0
= ϕπ πT +1
e
Aggregate demand (with yt+1
= 0) can then be written:
yT +1 = −rT +1 + ρT +1
|{z}
(225)
=0
= −ϕπ πT +1
We then make use of the aggregate supply (or Phillips curve) relationship:
e
πt = πt+1
+ λ yt
Recall that in the absence of productivity shocks natural ouput is zero, hence actual output
yt is equal to the output gap. We are going to assume (without loss of generality) that there
is perfect foresight and that inflation expectations are adaptive:
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e
πt+1
= πt+1 = πt
We now can write:
πT +1 = πT + λ yt+1
(226)
Plugging (226) into (225) allows to derive the T+1 system:
πT +1 =
1
πT
1 + λϕπ
(227)
ϕπ
πT
1 + λϕπ
(228)
yT +1 = −
Equilibrium at time T. Suppose first that the ZLB is not binding at time T. The
monetary policy rule reads:
rT = rT∗ + ϕπ πT
Aggregate demand:
yT = −rT + ρbT
while aggregate supply (since by assumption πT −1 = 0)
πT = λ y T
(229)
πT = −λrT + λb
ρT
(230)
Solving for the system at time T yields:
= −λrT∗ + λϕπ πT + λb
ρT
(231)
= λϕπ πT
(232)
The only solution to the above equation is πT = 0 which then implies yT = 0. Hence in a
normal regime (outside the ZLB), and in response to a negative discount factor (aggregate
demand) shock, monetary policy (which follows rule (222)) is able to keep inflation at zero
and the output gap at zero.
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It is interesting to ask whether there exists a limit to the size of the shock, i.e, to the
fall of output and inflation, before the ZLB starts binding. The lower limit is reached when
r∗ + ϕ π = rTmin = ζ − πT +1
|T {z π T}
rT
Substituting πT +1 from (227):
1
rT∗ + ϕπ πT = ζ −
πT
| {z }
1 + λϕπ
rT
Substituting rT∗ = ρbT
and solving for πT yields:
πTmin =
(ζ − ρbT )(1 + λϕπ )
1 + ϕπ (1 + λϕπ )
Notice that if the shock ρbT to the natural real interest rate is sufficiently large the lower
bound on inflation can also take a negative value, i.e., deflation.
ZLB binding. Suppose that at time T the zero lower bound is binding, so that rt =
rtmin . This happens if the shock ρT
is sufficiently large. Hence we have
yT = −(rTmin − ρbT )
= −ζ + πT +1 + ρbT
Hence a negative shock ρbT lowers output at time T. In addition, in a liquidity trap, expectations of future deflation also are contractionary. Let’s substitute πT +1 from (227) and
obtain:
yT = −ζ + πT +1 + ρbT
1
=
πT + ρbT − ζ
1 + λϕπ
Aggregate demand relationship at the ZLB. The above equation describes the
aggregate demand equation in a liquidity trap. The key novelty, which is specific to the
ZLB state, is that this equation now describes a positive relationship between output and
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Figure 34: Aggregate Demand curve in and out of a liquidity trap. At the zero lower bound
the slope of the AD curve becomes positive
inflation, a radical change of perspective. The new shape of the AD equation is represented
in Figure 34. The AD curve is (as usual) negatively sloped outside the liquidity trap, but
it becomes positively sloped at the zero lower bound. Hence the AD curve features a kink,
precisely at the π min point.
Deflationary spiral. A positively sloped AD equation at the ZLB can be the source
of a deflationary spiral. The reason is as follows. At the ZLB the real interest rate is
determined only by two terms: the term premium ζ and the inflation expectations. Since
low inflation in period T generates lower (expected) inflation in T + 1, the shock at time T
induces a rise in the real interest rate in period T, thereby decreasing aggregate demand. In
normal times (ie., outside the ZLB) the central bank would react by lowering the nominal
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interest rate to offset the shock and preventing the real interest rate from rising. But at the
ZLB this is unfeasible. Hence, with a binding ZLB, a deflationary shock at T is not only
contractionary per se; its effect is amplified via a rise in the real interest rate, induced by
lower inflation expectations.
The deflationary spiral that takes at the ZLB can be therefore described as follows
→
ρ shock
AD falls
→ πT falls (via AS)
πT +1 falls
→
real int. rate rises
→ AD falls further
Does the spiral eventually die out? One can show that the condition for the spiral to
settle down is that the slope of the AD curve be greater than the slope of the AS curve.
In Figure 35 we represent the possible equilibria, in and out of a liquidity trap. The
economy is initially in a normal equilibrium at point A. A negative aggregate demand shock
shifts the AD curve to the left. If that shock is sufficiently large (in absolute value) the new
equilibrium is at the liquidity trap point B. Notice that below point π min the slope of the
AD curve is greater than the one of the AS curve.
yT =
1
πT + ρbT − ζ
1 + λϕπ
πT = λ y T
Combining yields the solution:
yZLB =
πZLB =
ρbT − ζ
λ
1 − 1+λϕ
π
λ (b
ρT − ζ)
λ
1 − 1+λϕ
π
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Figure 35: Equilibrium in and out of a liquidity trap.
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Amplification at the ZLB. Notice that the elasticity of output to an AD shock at
the ZLB is
∂yZLB
∂ ρbT
=
=
1
λ
1 − 1+λϕ
π
1 + λϕπ
1 + λ(ϕπ − 1)
That elasticity is greater than the one outside the ZLB:
1
∂yT
=
∂ ρbT
1 + λϕπ
In other words, AD shocks that push the economy to the ZLB generate an amplified
contractionary effect on output. For two reasons. First, at the ZLB monetary policy becomes
impotent and cannot mitigate the impact of the shock. Second, at the ZLB the deflationary
spiral described above is set in motion.
23.1
Paradoxes at the zero lower bound
The liquidity trap is a state of the economy in which traditional macroeconomic mechanisms
are reversed. We illustrate this point via a series of paradoxes.
The paradox of flexibility. In the AS-AD system the degree of nominal price rigidity
determines the value of λ, i.e., the slope of the Phillips curve. Depending on whether the
equilibrium of the economy is inside or outside the liquidity trap, a higher price flexibility
(i.e., a larger value of λ) can either dampen or amplify the effect of an aggregate demand
shock. Figure 36 illustrates this point. The economy starts at the initial equilibrium in point
A. If the AD shock is not too large, so that the equilibrium remains in the normal regime,
′
the economy shifts to point B under low price flexibility, and to point B under high price
flexibility. The higher the degree of price flexibility, the smaller the contractionary effect on
output.
However if the shock is sufficiently large the economy plunges into a liquidity trap. The
′
equilibrium is either at point C (low price flexibility) or at point C . In this regime the
contractionary effect is larger under high price flexibility. In this situation those firms that
adjust their prices downward following the shock to aggregate demand tend to exacerbate the
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Figure 36: Paradox of flexibility. In a normal regime higher price flexibility helps dampening
the effect of a negative demand shock on output. In a liquidity trap higher flexibility amplifies
the contractionary effect.
deflationary spiral described above (lower demand leads to lower current and future inflation,
which in turn raises the real interest rate and depresses current demand even further).
Paradox of toil. Suppose that after a contractionary shock the economy finds it self
in a liquidity trap (so that the AD curve becomes upward sloping). What happens if, starting
from liqudity trap state, all agents in the economy desire to work more?
We can represent this phenomenon as an aggregate increase in labor supply, which shifts
the AS curve down to the right (see Figure 37). Starting from point A, the equilibrium
shifts to point B. Hence we see that the attempt by all agents to increase the supply of
labor (presumably as a way to try to address the recessionary effect of the shock) leads to a
contraction in aggregate activity. Trying to work more, in the aggregate, leads to a decline in
the level of output. In general, at the ZLB, any positive supply shock, due to its deflationary
effect, tends to generate lower current and future inflation, thereby raising the real interest
rate and depressing aggregate demand.
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Figure 37: Paradox of toil at the liqudity trap. An attempt by all agents in the economy to
increase the supply of labor leads to a contraction in output.
Paradox of thrift. What happens in a liquidity trap if all agents in the economy try
to save more? We can represent this event as an inward shift of the aggregate demand curve.
If aggregate savings fall for any given level of output also aggregate demand will fall. In a
liquidity trap, as shown in Figure 38 this shifts the equilibrium from point A to point B.
The result is a contraction in aggregate output. The reason is that the shock is deflationary,
and at the ZLB a fall in inflation raises the real interest rate depresses demand. If the fall
in output is large enough, in equilibrium, also aggregate savings (which are proportional to
output) might fall, even if the saving rate is higher.
Formally, savings S are share of aggregate output:
S =s·Y
where s is the marginal propensity to save. The increase in the desire to save can be
described as a rise in the marginal propensity to save s, holding constant Y. This per se
would induce a rise in aggregate savings. But in equilibrium, as shown in Figure 38, output
Y falls. If the fall in Y is large enough this will lead to final contraction in savings. The
paradox therefore is that if all agents desire to save more, in the aggregate this will lead to
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Figure 38: Paradox of thrift. In a liquidity trap a higher desire to save leads to a contraction
in aggregate output and in equilibrium to a fall in aggregate savings.
a fall in aggregate savings.
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